#+PROPERTY: header-args:sage :session result #+BEGIN_SRC sage :exports none from e8theta_degree3.hecke_module import (HalfIntMatElement, HeckeModule, spinor_l_euler_factor, rankin_convolution_degree1, from_spinor_to_standard_l) from e8theta_degree3.gl3_repn import gl3_repn_module from e8theta_degree3.results.data.data_utils import (data_dir, half_int_mat_to_list, sort_ts, dict_sum, gcd_of_dict_vals, modulo_p, factor_latex, factorization_normalized, _to_diag_mats, vec_dict_gcd) T0 = HalfIntMatElement(matrix([[1, 1 / 2, 1 / 2], [1 / 2, 1, 1 / 2], [1 / 2, 1 / 2, 1]])) T1 = HalfIntMatElement(diagonal_matrix([1, 1, 1])) i = QuadraticField(-1, name="i").gen() trans_mats = load(os.path.join(data_dir(), "trans_mats.sobj")) #+END_SRC #+RESULTS: * Polynomial representation The dimension of the polynomial representation of weight =(18, 17, 5)= is =195=. We sort SSYT of weight =(13, 12, 0)= as in [[file:SSYT.org][this file]]. * Basis of the space of cusp forms #+BEGIN_SRC sage :exports none dicts = load(os.path.join(data_dir(), "wt18_17_5_dicts.sobj")) S18_17_5 = HeckeModule(dicts, lin_indep_ts=[T0, T1]) #+END_SRC #+RESULTS: #+BEGIN_SRC sage :exports none mat0 = matrix(3, [-5, -3*i, 4, 6*i, -2, 0, 0, 0, 3, 5*i, 2, 0, 4, 2*i, 0, 0, 0, 6*i, 3, 3*i, 6, 0, 0, 0]) mat1 = matrix(3, [-7, -5*i, 6, 8*i, -2, 0, 0, 0, -8, -6*i, -3, i, 0, 6*i, 0, 0, -26, -10*i, -25, i, -13, 37*i, 0, 0]) mat2 = matrix(3, [-6, -4*i, 5, 7*i, -2, 0, 0, 0, -11, -9*i, 0, 4*i, 1, 5*i, 0, 0, -27, -43*i, 20, 4*i, 31, -15*i, 0, 0]) #+END_SRC #+RESULTS: #+BEGIN_SRC sage :exports none print latex(mat0) print latex(mat1) print latex(mat2) #+END_SRC #+RESULTS: #+begin_example \left(\begin{array}{rrrrrrrr} -5 & -3 \sqrt{-1} & 4 & 6 \sqrt{-1} & -2 & 0 & 0 & 0 \\ 3 & 5 \sqrt{-1} & 2 & 0 & 4 & 2 \sqrt{-1} & 0 & 0 \\ 0 & 6 \sqrt{-1} & 3 & 3 \sqrt{-1} & 6 & 0 & 0 & 0 \end{array}\right) \left(\begin{array}{rrrrrrrr} -7 & -5 \sqrt{-1} & 6 & 8 \sqrt{-1} & -2 & 0 & 0 & 0 \\ -8 & -6 \sqrt{-1} & -3 & \sqrt{-1} & 0 & 6 \sqrt{-1} & 0 & 0 \\ -26 & -10 \sqrt{-1} & -25 & \sqrt{-1} & -13 & 37 \sqrt{-1} & 0 & 0 \end{array}\right) \left(\begin{array}{rrrrrrrr} -6 & -4 \sqrt{-1} & 5 & 7 \sqrt{-1} & -2 & 0 & 0 & 0 \\ -11 & -9 \sqrt{-1} & 0 & 4 \sqrt{-1} & 1 & 5 \sqrt{-1} & 0 & 0 \\ -27 & -43 \sqrt{-1} & 20 & 4 \sqrt{-1} & 31 & -15 \sqrt{-1} & 0 & 0 \end{array}\right) #+end_example Let =F1=, =F2= and =F3= are theta series of weight =(18, 17, 5)= associated with the following three matrices respectively. Then =F1=, =F2= and =F3= spans the space of cusp forms. #+BEGIN_SRC sage :exports results print mat0 #+END_SRC #+RESULTS: : [ -5 -3*i 4 6*i -2 0 0 0] : [ 3 5*i 2 0 4 2*i 0 0] : [ 0 6*i 3 3*i 6 0 0 0] #+BEGIN_SRC sage :exports results print mat1 #+END_SRC #+RESULTS: : [ -7 -5*i 6 8*i -2 0 0 0] : [ -8 -6*i -3 i 0 6*i 0 0] : [ -26 -10*i -25 i -13 37*i 0 0] #+BEGIN_SRC sage :exports results print mat2 #+END_SRC #+RESULTS: : [ -6 -4*i 5 7*i -2 0 0 0] : [ -11 -9*i 0 4*i 1 5*i 0 0] : [ -27 -43*i 20 4*i 31 -15*i 0 0] #+BEGIN_SRC sage :exports none gcd_of_dict_vals(S18_17_5.basis[0]).factor() #+END_SRC #+RESULTS: : 2^20 * 3^5 * 5 * 7 Fourier coefficients of =F1/(2^20 * 3^5 * 5 * 7)= are given as in [[file:F1.csv][F1]]. # (progn (re-search-forward "|") (org-table-export "./F1.csv" "orgtbl-to-csv")) #+BEGIN_SRC sage :results table :exports none ts18_17_5 = sort_ts(S18_17_5.basis[0].keys()) [(half_int_mat_to_list(t), S18_17_5.basis[0][t].vector/(2^20 * 3^5 * 5 * 7)) for t in ts18_17_5] #+END_SRC #+RESULTS: | [1, 1, 1, 1, 1, 1] | (0, 0, 0, 0, -388869978684360, -486087473355450, 0, 486087473355450, 388869978684360, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -194434989342180, -194434989342180, -243043736677725, -243043736677725, 0, 0, 243043736677725, 243043736677725, 194434989342180, 194434989342180, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -92698348581855, -94506007017549, -189012014035098, -21899884672449, -71207795453696, 121010203908500, 108506895311275, 108506895311275, 121010203908500, -71207795453696, -21899884672449, -189012014035098, -94506007017549, -92698348581855, 0, 0, 0, 0, 0, 0, 0, 0, -92698348581855, 1807658435694, 0, 83249066023107, 128447162259409, 93192846255412, 129997448439434, -12503308597225, 12503308597225, -129997448439434, -93192846255412, -128447162259409, -83249066023107, 0, -1807658435694, 92698348581855, 0, 0, 0, 0, 0, 194434989342180, 189012014035098, 167112129362649, 128447162259409, 35254316003997, 0, -74486626586159, -80385619080945, -80385619080945, -74486626586159, 0, 35254316003997, 128447162259409, 167112129362649, 189012014035098, 194434989342180, 0, 0, 388869978684360, 194434989342180, -48608747335545, -71207795453696, -192217999362196, -129997448439434, -142500757036659, -80385619080945, 0, 0, 80385619080945, 142500757036659, 129997448439434, 192217999362196, 71207795453696, 48608747335545, -194434989342180, -388869978684360, -486087473355450, -243043736677725, -243043736677725, -108506895311275, 0, 12503308597225, 142500757036659, 74486626586159, 74486626586159, 142500757036659, 12503308597225, 0, -108506895311275, -243043736677725, -243043736677725, -486087473355450, 0, 0, 243043736677725, 121010203908500, 192217999362196, 93192846255412, -35254316003997, 35254316003997, -93192846255412, -192217999362196, -121010203908500, -243043736677725, 0, 0, 486087473355450, 243043736677725, 48608747335545, 21899884672449, -167112129362649, -83249066023107, -83249066023107, -167112129362649, 21899884672449, 48608747335545, 243043736677725, 486087473355450, -388869978684360, -194434989342180, -194434989342180, -94506007017549, -1807658435694, 1807658435694, 94506007017549, 194434989342180, 194434989342180, 388869978684360, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) | | [1, 1, 1, 0, 0, 0] | (0, 0, 0, -6999659616318480, 0, 7777399573687200, 0, -7777399573687200, 0, 6999659616318480, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11867944165368, 0, 0, -997065858095096, -1207736775154432, 0, 0, 1393676175149260, 1393676175149260, 0, 0, -1207736775154432, -997065858095096, 0, 0, 11867944165368, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6999659616318480, 0, 0, 997065858095096, -210670917059336, 0, 0, 0, -358918384754340, 0, 0, -358918384754340, 0, 0, 0, -210670917059336, 997065858095096, 0, 0, -6999659616318480, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7777399573687200, 0, 0, -1393676175149260, 0, 0, 0, 358918384754340, 358918384754340, 0, 0, 0, -1393676175149260, 0, 0, 7777399573687200, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -7777399573687200, 0, 0, 1207736775154432, 210670917059336, 0, 0, 210670917059336, 1207736775154432, 0, 0, -7777399573687200, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6999659616318480, 0, 0, -11867944165368, -11867944165368, 0, 0, 6999659616318480, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) | | [1, 1, 2, 0, 0, 0] | (0, 0, 0, 83995915395821760, 0, -139993192326369600, 0, 163325391047431200, 0, -251987746187465280, 0, 1539925115590065600, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -142415329984416, 0, 0, 15621190316862192, 8781153979102560, 0, 0, -39140995427273040, -4842574167845280, 0, 0, 50105634051842880, 36460674076972800, 0, 0, -209400996627548352, -124246831948789680, 0, 0, 977545529536799520, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 83995915395821760, 0, 0, -15621190316862192, -6840036337759632, 0, 0, 0, -18235332766047696, 0, 0, 51490671414750528, 34968540891135168, 0, 0, 75773620800276144, 399782661951279744, 0, 0, 644309853627688992, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -139993192326369600, 0, 0, 39140995427273040, 34298421259427760, 0, 0, -51490671414750528, -16522130523615360, 0, 0, 0, -390851202107001600, 0, 0, -568111000712872800, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 163325391047431200, 0, 0, -50105634051842880, -13644959974870080, 0, 0, -75773620800276144, 324009041151003600, 0, 0, 568111000712872800, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -251987746187465280, 0, 0, 209400996627548352, 85154164678758672, 0, 0, -644309853627688992, 0, 0, 0, 0, 0, 0, 1539925115590065600, 0, 0, -977545529536799520, 0, 0) | | [1, 1, 3, 1, 1, 1] | (0, 0, 0, 0, -40442477783173440, -66107896376341200, 69996596163184800, 291652484013270000, 102661674372671040, -755963238562395840, -1166609936053080000, 8212933949813683200, 92395506935403936000, 0, 0, 0, 0, 0, 0, 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#+BEGIN_SRC sage :exports none gcd_of_dict_vals(S18_17_5.basis[1]).factor() #+END_SRC #+RESULTS: : 2^5 * 3^4 * 5 * 7^2 Fourier coefficients of =F2/(2^5 * 3^4 * 5 * 7^2)= are given as in [[file:F2.csv][F2]]. # (progn (re-search-forward "|") (org-table-export "./F2.csv" "orgtbl-to-csv")) #+BEGIN_SRC sage :results table :exports none [(half_int_mat_to_list(t), S18_17_5.basis[1][t].vector/(2^5 * 3^4 * 5 * 7^2)) for t in ts18_17_5] #+END_SRC #+RESULTS: | [1, 1, 1, 1, 1, 1] | (0, 0, 0, 0, 438300550283902927246120, 547875687854878659057650, 0, -547875687854878659057650, -438300550283902927246120, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 219150275141951463623060, 219150275141951463623060, 273937843927439329528825, 273937843927439329528825, 0, 0, -273937843927439329528825, -273937843927439329528825, -219150275141951463623060, -219150275141951463623060, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 129478221064819164882075, 121516987667281791653857, 243033975334563583307714, -70337912151559688978003, 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5^3 Fourier coefficients of =F3/(2^8 * 3^2 * 5^3)= are given as in [[file:F3.csv][F3]]. # (progn (re-search-forward "|") (org-table-export "./F3.csv" "orgtbl-to-csv")) #+BEGIN_SRC sage :results table :exports none [(half_int_mat_to_list(t), S18_17_5.basis[2][t].vector/(2^8 * 3^2 * 5^3)) for t in ts18_17_5] #+END_SRC #+RESULTS: | [1, 1, 1, 1, 1, 1] | (0, 0, 0, 0, -738097079296302085391592, -922621349120377606739490, 0, 922621349120377606739490, 738097079296302085391592, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -369048539648151042695796, -369048539648151042695796, -461310674560188803369745, -461310674560188803369745, 0, 0, 461310674560188803369745, 461310674560188803369745, 369048539648151042695796, 369048539648151042695796, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -104427322983251080870923, -136466101719580857061713, -272932203439161714123426, 61668882767047517905851, -86440496258869620167920, 330224974144380624945940, 260738629861045941835415, 260738629861045941835415, 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-267173511023289004766074392005760, -52060112863600636903620884887680, 0, 0) | | [2, 2, 2, 0, 0, 0] | (0, 0, 0, 799401315177989486267818309632, 0, -629866558023937731048027704320, 0, 629866558023937731048027704320, 0, -799401315177989486267818309632, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 156329681969331837020250158592, 0, 0, -492869962562988286270232647168, -350253357632253352358312270848, 0, 0, 701677186206397282641309512960, 701677186206397282641309512960, 0, 0, -350253357632253352358312270848, -492869962562988286270232647168, 0, 0, 156329681969331837020250158592, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 799401315177989486267818309632, 0, 0, 492869962562988286270232647168, 142616604930734933911920376320, 0, 0, 0, -130665545459571536323813731072, 0, 0, -130665545459571536323813731072, 0, 0, 0, 142616604930734933911920376320, 492869962562988286270232647168, 0, 0, 799401315177989486267818309632, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -629866558023937731048027704320, 0, 0, -701677186206397282641309512960, 0, 0, 0, 130665545459571536323813731072, 130665545459571536323813731072, 0, 0, 0, -701677186206397282641309512960, 0, 0, -629866558023937731048027704320, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 629866558023937731048027704320, 0, 0, 350253357632253352358312270848, -142616604930734933911920376320, 0, 0, -142616604930734933911920376320, 350253357632253352358312270848, 0, 0, 629866558023937731048027704320, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -799401315177989486267818309632, 0, 0, -156329681969331837020250158592, -156329681969331837020250158592, 0, 0, -799401315177989486267818309632, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) | * Hecke eigenforms The characteristic polynomial of =T(2)= is given as follows. #+BEGIN_SRC sage :exports results S18_17_5.hecke_charpoly_tp(2).factor() #+END_SRC #+RESULTS: : (x - 285120) * (x^2 - 122112*x - 6040977408) Let =G1 = -27411940570812415758258813595452421519092205 * F1 + 426844081323506188348315458383164538880 * F2 + 1108258366195376623065993820933390921728 * F3= . Then it is an eigenform whose eigenvalue is equal to =285120=. #+BEGIN_SRC sage :exports none f18_17_5_0 = dict_sum((-27411940570812415758258813595452421519092205, 426844081323506188348315458383164538880, 1108258366195376623065993820933390921728), S18_17_5.basis) #+END_SRC #+RESULTS: The euler factor of the spinor L function at =2= is given as follows. #+BEGIN_SRC sage :exports results spl2_18_17_5_0 = spinor_l_euler_factor(2, f18_17_5_0) spl2_18_17_5_0.factor() #+END_SRC #+RESULTS: : (-1 + 131072*t)^2 * (1 + 75520*t + 17179869184*t^2) * (1 - 98496*t - 3084910592*t^2 - 1692148395147264*t^3 + 295147905179352825856*t^4) This satisfies conjecture 7.7 (i) at 2 for =f= of weight =16=, =g= of weight =20= and =h= of weight =16=. #+BEGIN_SRC sage :exports none K18_17_5 = NumberField(x^2 - 122112*x - 6040977408, names="a") #+END_SRC #+RESULTS: Let =a= be a root of =x^2 - 122112*x - 6040977408=. And let G2= (115755188162636478306143242492773391279012500590415297895*a - 280871869782212043390032985770730420043620758653959144573711360) * F1 + (-3520271549093870111699822711292893500461035796234240*a + 1792446151032915388102568677460511010394423622703062712320) * F2 + (-23730541121480814714846359164016657918248298142879744*a + 8813953574896657677285798006856812369943698972051863764992) * F3=. Then =G2= is an eigenform whose eigenvalue is equal to =a=. #+BEGIN_SRC sage :exports none f18_17_5_1 = dict_sum((115755188162636478306143242492773391279012500590415297895*K18_17_5.gen() - 280871869782212043390032985770730420043620758653959144573711360, -3520271549093870111699822711292893500461035796234240*K18_17_5.gen() + 1792446151032915388102568677460511010394423622703062712320, -23730541121480814714846359164016657918248298142879744*K18_17_5.gen() + 8813953574896657677285798006856812369943698972051863764992), S18_17_5.basis) #+END_SRC #+RESULTS: #+BEGIN_SRC sage :exports none c18_17_5_0 = vec_dict_gcd(f18_17_5_0); factor(c18_17_5_0) c18_17_5_1 = vec_dict_gcd(f18_17_5_1); factor(c18_17_5_1) #+END_SRC #+RESULTS: : 2^24 * 3^8 * 5^3 * 7^3 * 13 * 19 * 61 * 4723 * 53368181 * 30285838023276412271060640124610423 : 2^26 * 3^8 * 5^2 * 7^3 * 19 * 61 * 4723 * 53368181 * 30285838023276412271060640124610423 # (progn (re-search-forward "|") (org-table-export "./G1.csv" "orgtbl-to-csv")) #+BEGIN_SRC sage :results table :exports none [(half_int_mat_to_list(t), f18_17_5_0[t].vector/c18_17_5_0) for t in ts18_17_5] #+END_SRC #+RESULTS: | [1, 1, 1, 1, 1, 1] | (0, 0, 0, 0, -40, -50, 0, 50, 40, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -20, -20, -25, -25, 0, 0, 25, 25, 20, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 45, 23, 46, 11, 8, -20, -25, -25, -20, 8, 11, 46, 23, 45, 0, 0, 0, 0, 0, 0, 0, 0, 45, 22, 0, -37, -63, -60, -62, -5, 5, 62, 60, 63, 37, 0, -22, -45, 0, 0, 0, 0, 0, 20, -46, -35, -63, -3, 0, 69, 75, 75, 69, 0, -3, -63, -35, -46, 20, 0, 0, 40, 20, -5, 8, 28, 62, 57, 75, 0, 0, -75, -57, -62, -28, -8, 5, -20, -40, -50, -25, -25, 25, 0, 5, -57, -69, -69, -57, 5, 0, 25, -25, -25, -50, 0, 0, 25, -20, -28, -60, 3, -3, 60, 28, 20, -25, 0, 0, 50, 25, 5, -11, 35, 37, 37, 35, -11, 5, 25, 50, -40, -20, -20, 23, -22, 22, -23, 20, 20, 40, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) | | [1, 1, 1, 0, 0, 0] | (0, 0, 0, -720, 0, 800, 0, -800, 0, 720, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -360, 0, 0, 1960, 1280, 0, 0, -1540, -1540, 0, 0, 1280, 1960, 0, 0, -360, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -720, 0, 0, -1960, -680, 0, 0, 0, -180, 0, 0, -180, 0, 0, 0, -680, -1960, 0, 0, -720, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 800, 0, 0, 1540, 0, 0, 0, 180, 180, 0, 0, 0, 1540, 0, 0, 800, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -800, 0, 0, -1280, 680, 0, 0, 680, -1280, 0, 0, -800, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 720, 0, 0, 360, 360, 0, 0, 720, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) | | [1, 1, 2, 0, 0, 0] | (0, 0, 0, 8640, 0, -14400, 0, 16800, 0, -25920, 0, 158400, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4320, 0, 0, -14544, -11808, 0, 0, 25200, 17760, 0, 0, -12864, 37632, 0, 0, 187200, -77040, 0, 0, 174240, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8640, 0, 0, 14544, 2736, 0, 0, 0, 7152, 0, 0, -44736, -39744, 0, 0, 36720, 185472, 0, 0, -21600, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -14400, 0, 0, -25200, -7440, 0, 0, 44736, 4992, 0, 0, 0, -138240, 0, 0, -4320, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16800, 0, 0, 12864, 50496, 0, 0, -36720, 148752, 0, 0, 4320, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -25920, 0, 0, -187200, -264240, 0, 0, 21600, 0, 0, 0, 0, 0, 0, 158400, 0, 0, -174240, 0, 0) | | [1, 1, 3, 1, 1, 1] | (0, 0, 0, 0, -4160, -6800, 7200, 30000, 10560, -77760, -120000, 844800, 9504000, 0, 0, 0, 0, 0, 0, 0, 0, -2080, -2080, -3400, -3400, 3600, 3600, 15000, 15000, 5280, 5280, -38880, -38880, -60000, -60000, 422400, 422400, 4752000, 4752000, 0, 0, 0, 0, 0, 4680, 2392, 4784, 3672, -6080, -720, -27800, -2760, -32160, -9648, 17544, -39888, 164640, -13440, 4080, 200880, 603240, 2205720, 638880, 0, 0, 0, 4680, 2288, 0, 344, -11800, -5872, -23200, 8280, 5960, 14928, 71520, -41736, 135600, -74112, -261120, 60960, 445800, 882480, -1086720, 0, 0, 2080, -4784, -1112, -11800, -5928, 0, 23576, 47720, 28520, 65016, -61392, -76800, -154928, -576200, -184144, 39360, -381600, -1041120, 4160, 2080, -1320, -6080, -5360, 23200, 31480, 47720, 19200, 0, -96200, -261840, -174320, -500960, -313880, 39480, -891040, 268800, -6800, -3400, -7000, 27800, 25040, 5960, -8968, -65016, -126408, -261840, -87520, 0, -145720, 378360, -272040, 1021200, -7200, -3600, 11400, -32160, -22512, -71520, -113256, -76800, 78128, 500960, 187080, 378360, 650400, 0, 30000, 15000, 9720, -17544, -57432, 135600, 209712, 576200, 392056, 39480, 930520, -1021200, -10560, -5280, -44160, 164640, 178080, 261120, 322080, 39360, 420960, -268800, -77760, -38880, 21120, -4080, 196800, 445800, -436680, 1041120, 120000, 60000, 482400, 603240, -1602480, 1086720, 844800, 422400, -4329600, -638880, -9504000, -4752000) | | [2, 2, 2, 2, 2, 2] | (0, -38776320, -26173440, -5644800, -3704320, -5713280, 0, 5713280, 3704320, 5644800, 26173440, 38776320, 0, 0, 0, -19388160, -19388160, -13086720, -13086720, -2822400, -2822400, -1852160, -1852160, -2856640, -2856640, 0, 0, 2856640, 2856640, 1852160, 1852160, 2822400, 2822400, 13086720, 13086720, 19388160, 19388160, 0, 0, -38776320, -19388160, -6301440, 0, 3962880, 13717440, 7956800, 11950720, 2396480, 1867520, -5480960, -6232000, -6232000, -5480960, 1867520, 2396480, 11950720, 7956800, 13717440, 3962880, 0, -6301440, -19388160, -38776320, 26173440, 13086720, 10264320, 13717440, 5760640, 0, -9350080, -13872960, -13482240, -13704320, -751040, 751040, 13704320, 13482240, 13872960, 9350080, 0, -5760640, -13717440, -10264320, -13086720, -26173440, -5644800, -2822400, -970240, -11950720, -9554240, -13872960, -390720, 0, 15610560, 16472640, 16472640, 15610560, 0, -390720, -13872960, -9554240, -11950720, -970240, -2822400, -5644800, 3704320, 1852160, -1004480, 1867520, 7348480, 13704320, 12953280, 16472640, 0, 0, -16472640, -12953280, -13704320, -7348480, -1867520, 1004480, -1852160, -3704320, -5713280, -2856640, -2856640, 6232000, 0, 751040, -12953280, -15610560, -15610560, -12953280, 751040, 0, 6232000, -2856640, -2856640, -5713280, 0, 0, 2856640, -5480960, -7348480, -13482240, 390720, -390720, 13482240, 7348480, 5480960, -2856640, 0, 0, 5713280, 2856640, 1004480, -2396480, 9554240, 9350080, 9350080, 9554240, -2396480, 1004480, 2856640, 5713280, -3704320, -1852160, 970240, 7956800, -5760640, 5760640, -7956800, -970240, 1852160, 3704320, 5644800, 2822400, -10264320, -3962880, -3962880, -10264320, 2822400, 5644800, -26173440, -13086720, 6301440, -6301440, 13086720, 26173440, 38776320, 19388160, 19388160, 38776320, 0, 0) | | [1, 3, 3, 2, 0, 0] | (0, -7603200, -1843200, 967680, -6753280, -9286400, 0, 9286400, 6753280, -967680, 1843200, 7603200, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -203005440, -44478720, -75098880, -14515200, -16819200, -7237440, 10049856, 9634688, 35428608, 24467136, 42420800, 34780800, 34780800, 42420800, 24467136, 35428608, 9634688, 10049856, -7237440, -16819200, -14515200, -75098880, -44478720, -203005440, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 155727360, 61421760, 27125184, 7097472, -30718848, 2844032, -30208128, 17607552, -8112768, 18935424, 18935424, -8112768, 17607552, -30208128, 2844032, -30718848, 7097472, 27125184, 61421760, 155727360, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -166379520, -185311680, 4877760, -89440640, 35672960, -41442560, 1956096, -4597632, -4597632, 1956096, -41442560, 35672960, -89440640, 4877760, -185311680, -166379520, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 746762880, 782910400, -305295936, 37850112, -348424448, -285673728, -285673728, -348424448, 37850112, -305295936, 782910400, 746762880, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1644082560, -442555200, 1867103040, 13962240, 13962240, 1867103040, -442555200, -1644082560, 0, 0, 0, 0, 0, 0, 18422954880, 754300800, 754300800, 18422954880, 0, 0) | | [2, 2, 2, 0, 0, 0] | (0, 0, 0, -203581440, 0, 226595840, 0, -226595840, 0, 203581440, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 62092800, 0, 0, 190024192, 101123072, 0, 0, -181062400, -181062400, 0, 0, 101123072, 190024192, 0, 0, 62092800, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -203581440, 0, 0, -190024192, -88901120, 0, 0, 0, -40741632, 0, 0, -40741632, 0, 0, 0, -88901120, -190024192, 0, 0, -203581440, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 226595840, 0, 0, 181062400, 0, 0, 0, 40741632, 40741632, 0, 0, 0, 181062400, 0, 0, 226595840, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -226595840, 0, 0, -101123072, 88901120, 0, 0, 88901120, -101123072, 0, 0, -226595840, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 203581440, 0, 0, -62092800, -62092800, 0, 0, 203581440, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) | # (progn (re-search-forward "|") (org-table-export "./G2.csv" "orgtbl-to-csv")) #+BEGIN_SRC sage :results table :exports none [(half_int_mat_to_list(t), f18_17_5_1[t].vector/c18_17_5_1) for t in ts18_17_5] #+END_SRC #+RESULTS: | [1, 1, 1, 1, 1, 1] | (0, 0, 0, 0, 109660723214832720*a - 11993231335354332456960, 137075904018540900*a - 14991539169192915571200, 0, -137075904018540900*a + 14991539169192915571200, -109660723214832720*a + 11993231335354332456960, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 54830361607416360*a - 5996615667677166228480, 54830361607416360*a - 5996615667677166228480, 68537952009270450*a - 7495769584596457785600, 68537952009270450*a - 7495769584596457785600, 0, 0, -68537952009270450*a + 7495769584596457785600, -68537952009270450*a + 7495769584596457785600, -54830361607416360*a + 5996615667677166228480, -54830361607416360*a + 5996615667677166228480, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9835533802319010*a + 3427585326941804536320, 16867392602874678*a + 857228062629649476096, 33734785205749356*a + 1714456125259298952192, -8586696626310912*a + 3931591022375514345216, 11898625593423982*a + 596837083722193569024, 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24466133001280996846592*a - 3606017124874398191327379456, 0, 0, 24466133001280996846592*a - 3606017124874398191327379456, 0, 0, 0, -2464336792962628462592*a + 4400100102405508995225747456, -39353278470705823002624*a + 472476935002975181007224832, 0, 0, -69839784984761208281088*a + 19941598133155865320341110784, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 43927312253196127365120*a - 14547359075981961354317660160, 0, 0, 70388510724950689653760*a - 3212189526204100746993991680, 0, 0, 0, -24466133001280996846592*a + 3606017124874398191327379456, -24466133001280996846592*a + 3606017124874398191327379456, 0, 0, 0, 70388510724950689653760*a - 3212189526204100746993991680, 0, 0, 43927312253196127365120*a - 14547359075981961354317660160, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -43927312253196127365120*a + 14547359075981961354317660160, 0, 0, -36888941677743194540032*a - 3927623167402533814218522624, 2464336792962628462592*a - 4400100102405508995225747456, 0, 0, 2464336792962628462592*a - 4400100102405508995225747456, -36888941677743194540032*a - 3927623167402533814218522624, 0, 0, -43927312253196127365120*a + 14547359075981961354317660160, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 69839784984761208281088*a - 19941598133155865320341110784, 0, 0, 39722418694352209299456*a + 7790039123338503295721275392, 39722418694352209299456*a + 7790039123338503295721275392, 0, 0, 69839784984761208281088*a - 19941598133155865320341110784, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) | The euler factor of the spinor L function at =2= is given as follows. #+BEGIN_SRC sage :exports results spl2_18_17_5_1 = spinor_l_euler_factor(2, f18_17_5_1) spl2_18_17_5_1.factor() #+END_SRC #+RESULTS: : (1 - 131072*t)^2 * (1 + (-a + 262144)*t + (-36352*a + 24053547008)*t^2 + (8824815616*a + 1010090408673280)*t^3 + (-624522604576768*a + 413236791008634601472)*t^4 + (-295147905179352825856*a + 77371252455336267181195264)*t^5 + 5070602400912917605986812821504*t^6) Fourier coefficients of =G1/(2^24 * 3^8 * 5^3 * 7^3 * 13 * 19 * 61 * 4723 * 53368181 * 30285838023276412271060640124610423)= is given in [[file:G1.csv]]. Fourier coefficients of =G2/(2^26 * 3^8 * 5^2 * 7^3 * 19 * 61 * 4723 * 53368181 * 30285838023276412271060640124610423)= is given in [[file:G2.csv]]. #+BEGIN_SRC sage :exports none f12 = CuspForms(1, 12).basis()[0] f16 = CuspForms(1, 16).basis()[0] f20 = CuspForms(1, 20).basis()[0] rankin_convolution_degree1(f16, f20, 2).factor() rankin_convolution_degree1(f16, f16, 2, a=2).factor() #+END_SRC #+RESULTS: : 1 - 98496*t - 3084910592*t^2 - 1692148395147264*t^3 + 295147905179352825856*t^4 : (-1 + 131072*t)^2 * (1 + 75520*t + 17179869184*t^2) #+BEGIN_SRC sage :exports none factorization_normalized(from_spinor_to_standard_l(spl2_18_17_5_0)) factorization_normalized(from_spinor_to_standard_l(spl2_18_17_5_1)) #+END_SRC #+RESULTS: : (1 - t) * (1 + 295/512*t + t^2) * (1 - 1539/2048*t - 1471/8192*t^2 - 1539/2048*t^3 + t^4) : (1 - t) * (1 + (-1/131072*a + 2)*t + (-71/33554432*a + 91757/65536)*t^2 + (263/67108864*a + 58795/131072)*t^3 + (-71/33554432*a + 91757/65536)*t^4 + (-1/131072*a + 2)*t^5 + t^6) #+BEGIN_SRC sage :exports none print factor_latex(spl2_18_17_5_0) spl2_18_17_5_1.factor() print factor_latex(from_spinor_to_standard_l(spl2_18_17_5_0)) from_spinor_to_standard_l(spl2_18_17_5_1).factor() #+END_SRC #+RESULTS: : \left(1 - 2^{17} t\right)^{2}\left(1 + 2^{8} \cdot 5 \cdot 59 t + 2^{34} t^{2}\right)\left(1 - 2^{6} \cdot 3^{4} \cdot 19 t - 2^{21} \cdot 1471 t^{2} - 2^{40} \cdot 3^{4} \cdot 19 t^{3} + 2^{68} t^{4}\right) : (1 - 131072*t)^2 * (1 + (-a + 262144)*t + (-36352*a + 24053547008)*t^2 + (8824815616*a + 1010090408673280)*t^3 + (-624522604576768*a + 413236791008634601472)*t^4 + (-295147905179352825856*a + 77371252455336267181195264)*t^5 + 5070602400912917605986812821504*t^6) : \left(1 - t\right)\left(1 + 2^{-9} \cdot 5 \cdot 59 t + t^{2}\right)\left(1 - 2^{-11} \cdot 3^{4} \cdot 19 t - 2^{-13} \cdot 1471 t^{2} - 2^{-11} \cdot 3^{4} \cdot 19 t^{3} + t^{4}\right) : (1 - t) * (1 + (-1/131072*a + 2)*t + (-71/33554432*a + 91757/65536)*t^2 + (263/67108864*a + 58795/131072)*t^3 + (-71/33554432*a + 91757/65536)*t^4 + (-1/131072*a + 2)*t^5 + t^6) #+BEGIN_SRC sage :exports none _R. = PolynomialRing(ZZ, 1, order="neglex") __S = PolynomialRing(_R, 1, order="neglex", names="t") _S. = PolynomialRing(QQ, 1, order="neglex") print factor_latex((1 - 131072*t)^2) _spl2_18_17_5_1 = (1 + (-omega + 262144)*__S.gen() + (-36352*omega + 24053547008)*__S.gen()^2 + (8824815616*omega + 1010090408673280)*__S.gen()^3 + (-624522604576768*omega + 413236791008634601472)*__S.gen()^4 + (-295147905179352825856*omega + 77371252455336267181195264)*__S.gen()^5 + 5070602400912917605986812821504*__S.gen()^6) print "+".join(["%s %s" % (latex(_spl2_18_17_5_1[j].factor()), latex(t^j)) for j in range(7)]) #+END_SRC #+RESULTS: : \left(1 - 2^{17} t\right)^{2} : 1 1+\left(-1\right) \cdot (-262144 + \omega) t+\left(-1\right) \cdot (-46979584 + 71 \omega) \cdot 2^{9} t^{2}+2^{25} \cdot (30103040 + 263 \omega) t^{3}+\left(-1\right) \cdot (-46979584 + 71 \omega) \cdot 2^{43} t^{4}+\left(-1\right) \cdot (-262144 + \omega) \cdot 2^{68} t^{5}+2^{102} t^{6} #+BEGIN_SRC sage :exports none _R. = PolynomialRing(QQ, 1, order=TermOrder('neglex')) __S = PolynomialRing(_R, 1, order=TermOrder('neglex'), names="t") _spl2_18_17_5_1_std = (1 + (-1/131072*omega + 2)*__S.gen() + (-71/33554432*omega + 91757/65536)*__S.gen()^2 + (263/67108864*omega + 58795/131072)*__S.gen()^3 + (-71/33554432*omega + 91757/65536)*__S.gen()^4 + (-1/131072*omega + 2)*__S.gen()^5 + __S.gen()^6) for j in range(7): print _spl2_18_17_5_1_std[j].factor() for j in range(1, 6): print latex((_spl2_18_17_5_1_std[j]//_spl2_18_17_5_1_std[j].factor()[0][0]).factor()) #+END_SRC #+RESULTS: #+begin_example 1 (-1/131072) * (-262144 + omega) (-1/33554432) * (-46979584 + 71*omega) (1/67108864) * (30103040 + 263*omega) (-1/33554432) * (-46979584 + 71*omega) (-1/131072) * (-262144 + omega) 1 -1 \cdot 2^{-17} -1 \cdot 2^{-25} 2^{-26} -1 \cdot 2^{-25} -1 \cdot 2^{-17} #+end_example #+BEGIN_SRC sage :exports none latex((115755188162636478306143242492773391279012500590415297895 * omega - 280871869782212043390032985770730420043620758653959144573711360).factor()) #+END_SRC #+RESULTS: : (-1146415795029436911796053003145838449157635749607996508464128 + 472470155765863176759768336705197515424540818736388971 \omega) \cdot 5 \cdot 7^{2} * Congruence Fourier coefficients and Hecke eigenvalues of =G1= and =670 * G2= are congruent modulo a prime ideal over =757=. #+BEGIN_SRC sage :exports none def _check_cong_at(t): return all(modulo_p(a, 757, 285120) == modulo_p(b * 670, 757, 285120) for a, b in zip(f18_17_5_0[t].vector, f18_17_5_1[t].vector)) all(_check_cong_at(t) for t in f18_17_5_0) #+END_SRC #+RESULTS: : True #+BEGIN_SRC sage trans_mats = load(os.path.join(data_dir(), "trans_mats.sobj")) trans_mats[(18, 17, 5)].change_ring(FiniteField(757)).det() != 0 #+END_SRC #+RESULTS: : True #+BEGIN_SRC sage :exports none f18_17_5_0[T0].vector.change_ring(FiniteField(757)) #+END_SRC #+RESULTS: : (0, 0, 0, 0, 220, 275, 0, 482, 537, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 110, 110, 516, 516, 0, 0, 241, 241, 647, 647, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 131, 252, 504, 318, 713, 110, 516, 516, 110, 713, 318, 504, 252, 131, 0, 0, 0, 0, 0, 0, 0, 0, 131, 636, 0, 582, 725, 330, 341, 406, 351, 416, 427, 32, 175, 0, 121, 626, 0, 0, 0, 0, 0, 647, 253, 571, 725, 395, 0, 756, 723, 723, 756, 0, 395, 725, 571, 253, 647, 0, 0, 537, 647, 406, 713, 603, 416, 65, 723, 0, 0, 34, 692, 341, 154, 44, 351, 110, 220, 275, 516, 516, 241, 0, 351, 692, 1, 1, 692, 351, 0, 241, 516, 516, 275, 0, 0, 241, 110, 154, 330, 362, 395, 427, 603, 647, 516, 0, 0, 482, 241, 351, 439, 186, 175, 175, 186, 439, 351, 241, 482, 220, 110, 110, 252, 121, 636, 505, 647, 647, 537, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) * Transformation matrix The matrix =B_{(18, 17, 5)}= is the block diagonal matrix of the following matrices. #+BEGIN_SRC sage :exports results for _a in _to_diag_mats((18, 17, 5), trans_mats[(18, 17, 5)]): print _a print "" #+END_SRC #+RESULTS: #+begin_example [1] [12] [66] [220] [495] [792] [924] [792] [495] [220] [66] [12] [1] [1] [ 13 -12] [-12 24] [ 144 -132] [-132 198] [ 726 -660] [-660 880] [ 2200 -1980] [-1980 2475] [ 4455 -3960] [-3960 4752] [ 6336 -5544] [-5544 6468] [ 6468 -5544] [-5544 6336] [ 4752 -3960] [-3960 4455] [ 2475 -1980] [-1980 2200] [ 880 -660] [-660 726] [ 198 -132] [-132 144] [ 24 -12] [-12 13] [1] [12] [ 78 -66] [-66 198] [ 792 -660] [-660 1320] [ 3630 -2970] [-2970 4950] [ 9900 -7920] [-7920 11880] [ 17820 -13860] [-13860 19404] [ 22176 -16632] [-16632 22176] [ 19404 -13860] [-13860 17820] [11880 -7920] [-7920 9900] [ 4950 -2970] [-2970 3630] [1320 -660] [-660 792] [198 -66] [-66 78] [12] [66] [ 286 -220] [-220 880] [ 2640 -1980] [-1980 4950] [10890 -7920] [-7920 15840] [ 26400 -18480] [-18480 32340] [ 41580 -27720] [-27720 44352] [ 44352 -27720] [-27720 41580] [ 32340 -18480] [-18480 26400] [15840 -7920] [-7920 10890] [ 4950 -1980] [-1980 2640] [ 880 -220] [-220 286] [66] [220] [ 715 -495] [-495 2475] [ 5940 -3960] [-3960 11880] [ 21780 -13860] [-13860 32340] [ 46200 -27720] [-27720 55440] [ 62370 -34650] [-34650 62370] [ 55440 -27720] [-27720 46200] [ 32340 -13860] [-13860 21780] [11880 -3960] [-3960 5940] [2475 -495] [-495 715] [220] [495] [1287 -792] [-792 4752] [ 9504 -5544] [-5544 19404] [ 30492 -16632] [-16632 44352] [ 55440 -27720] [-27720 62370] [ 62370 -27720] [-27720 55440] [ 44352 -16632] [-16632 30492] [19404 -5544] [-5544 9504] [4752 -792] [-792 1287] [495] [792] [1716 -924] [-924 6468] [11088 -5544] [-5544 22176] [ 30492 -13860] [-13860 41580] [ 46200 -18480] [-18480 46200] [ 41580 -13860] [-13860 30492] [22176 -5544] [-5544 11088] [6468 -924] [-924 1716] [792] [924] [1716 -792] [-792 6336] [ 9504 -3960] [-3960 17820] [21780 -7920] [-7920 26400] [26400 -7920] [-7920 21780] [17820 -3960] [-3960 9504] [6336 -792] [-792 1716] [924] [792] [1287 -495] [-495 4455] [ 5940 -1980] [-1980 9900] [10890 -2970] [-2970 10890] [ 9900 -1980] [-1980 5940] [4455 -495] [-495 1287] [792] [495] [ 715 -220] [-220 2200] [2640 -660] [-660 3630] [3630 -660] [-660 2640] [2200 -220] [-220 715] [495] [220] [286 -66] [-66 726] [ 792 -132] [-132 792] [726 -66] [-66 286] [220] [66] [ 78 -12] [-12 144] [144 -12] [-12 78] [66] [12] [13 -1] [-1 13] [12] [1] [1] #+end_example