// Make newform 35.5.c.e in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_35_c();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_35_c_Hecke();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_35_5_c_e();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_35_5_c_e();". // This may take a long time! To see verbose output, uncomment the SetVerbose line below. // The default sign is -1. You can change this with the optional parameter "sign". function ConvertToHeckeField(input: pass_field := false, Kf := []) if not pass_field then poly := [4177936, 0, 190880, 0, 7113, 0, 110, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [188720, 0, 12513, 0, 125, 0, 2, 0], [47852, 0, -4064, 0, -125, 0, -2, 0], [0, -1118204, 0, -141325, 0, -2166, 0, -25], [0, 6052420, 0, 141325, 0, 2166, 0, 25], [517664, 0, 22497, 0, 436, 0, 3, 0], [-13776560, 26249794, -913449, 330432, -9125, 783, -146, -113], [-13776560, 11977428, -913449, 782451, -9125, 16867, -146, 154]]; Rf_basisdens := [1, 16898, 8449, 4934216, 4934216, 1988, 2467108, 2467108]; Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf); Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]]; inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf); return Eltseq(inp_vec); end function; // To make the character of type GrpDrchElt, type "MakeCharacter_35_c();" function MakeCharacter_35_c() N := 35; order := 2; char_gens := [22, 31]; v := [1, 1]; // chi(gens[i]) = zeta^v[i] assert UnitGenerators(DirichletGroup(N)) eq char_gens; F := CyclotomicField(order); chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]); return MinimalBaseRingCharacter(chi); end function; // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_35_c_Hecke();" function MakeCharacter_35_c_Hecke(Kf) N := 35; order := 2; char_gens := [22, 31]; char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0]]; assert UnitGenerators(DirichletGroup(N)) eq char_gens; values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field F := Universe(values);// the Hecke field chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values); return chi; end function; function ExtendMultiplicatively(weight, aps, character) prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec primes := PrimesUpTo(prec); prime_powers := primes; assert #primes eq #aps; log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1) F := Universe(aps); FXY := PolynomialRing(F, 2); // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ... R := PowerSeriesRing(FXY : Precision := log_prec + 1); recursion := Coefficients(1/(1 - X*T + Y*T^2)); coeffs := [F!0: i in [1..(prec+1)]]; coeffs[1] := 1; //a_1 for i := 1 to #primes do p := primes[i]; coeffs[p] := aps[i]; b := p^(weight - 1) * F!character(p); r := 2; p_power := p * p; //deals with powers of p while p_power le prec do Append(~prime_powers, p_power); coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]); p_power *:= p; r +:= 1; end while; end for; Sort(~prime_powers); for pp in prime_powers do for k := 1 to Floor(prec/pp) do if GCD(k, pp) eq 1 then coeffs[pp*k] := coeffs[pp]*coeffs[k]; end if; end for; end for; return coeffs; end function; function qexpCoeffs() // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" weight := 5; raw_aps := [[0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 5, 0, 0, -1, 1], [-44, 0, -6, 0, 0, 0, 0, 0], [0, 3, 0, 28, 0, 0, 0, 6], [0, -2, 0, 100, 0, 0, 0, -4], [0, -17, 0, 0, 0, -2, 0, 0], [0, -5, 0, 5, 94, 0, -10, 0], [-6, 0, -4, 0, 0, 0, 0, 0], [0, -20, 0, 0, 0, 8, 0, 0], [0, -10, 0, 10, -134, 0, -20, 0], [0, -36, 0, 0, 0, 2, 0, 0], [0, 25, 0, -25, -56, 0, 50, 0], [0, 3, 0, 209, 0, 0, 0, 6], [0, 6, 0, -6, -90, 0, 12, 0], [0, -61, 0, 0, 0, -26, 0, 0], [0, -35, 0, 0, 0, 36, 0, 0], [0, 5, 0, -5, 716, 0, 10, 0], [-2344, 0, 184, 0, 0, 0, 0, 0], [0, -112, 0, 956, 0, 0, 0, -224], [-976, 0, -204, 0, 0, 0, 0, 0], [0, 6, 0, -109, 0, 0, 0, 12], [0, 258, 0, 0, 0, 60, 0, 0], [0, -138, 0, -2060, 0, 0, 0, -276], [0, 309, 0, 0, 0, -2, 0, 0], [0, 289, 0, 1089, 0, 0, 0, 578], [0, 19, 0, -19, -220, 0, 38, 0], [-1496, 0, -574, 0, 0, 0, 0, 0], [0, 10, 0, -10, 1294, 0, 20, 0], [0, -135, 0, 135, -2994, 0, -270, 0], [0, -831, 0, 0, 0, 26, 0, 0], [0, 124, 0, -124, 2100, 0, 248, 0], [0, -305, 0, 0, 0, -106, 0, 0], [11704, 0, 906, 0, 0, 0, 0, 0], [25616, 0, 534, 0, 0, 0, 0, 0], [0, -477, 0, -2528, 0, 0, 0, -954], [0, -29, 0, 29, 4820, 0, -58, 0], [0, -225, 0, 10223, 0, 0, 0, -450], [0, 245, 0, 10704, 0, 0, 0, 490], [-1372, 0, 3130, 0, 0, 0, 0, 0], [0, -1853, 0, 0, 0, -158, 0, 0], [-25128, 0, -2620, 0, 0, 0, 0, 0], [0, 410, 0, -410, -4966, 0, 820, 0], [0, -430, 0, 430, -1814, 0, -860, 0], [0, -1426, 0, 0, 0, -172, 0, 0], [-7348, 0, -1770, 0, 0, 0, 0, 0], [0, -21, 0, -7175, 0, 0, 0, -42], [0, -1110, 0, -4219, 0, 0, 0, -2220], [0, -1081, 0, 0, 0, -54, 0, 0], [0, 320, 0, -320, -1316, 0, 640, 0], [-48192, 0, 2090, 0, 0, 0, 0, 0], [0, 1610, 0, 0, 0, 34, 0, 0], [0, -1511, 0, 0, 0, 658, 0, 0], [0, -294, 0, -27016, 0, 0, 0, -588], [0, -645, 0, 645, -8636, 0, -1290, 0], [0, -3211, 0, 0, 0, -40, 0, 0], [0, 412, 0, 0, 0, -464, 0, 0], [0, -746, 0, 746, 7570, 0, -1492, 0], [16576, 0, 3654, 0, 0, 0, 0, 0], [0, 132, 0, 14755, 0, 0, 0, 264], [0, 575, 0, 5040, 0, 0, 0, 1150], [0, -226, 0, 31901, 0, 0, 0, -452], [0, 1086, 0, 0, 0, 68, 0, 0], [0, 976, 0, -12960, 0, 0, 0, 1952], [0, 430, 0, -430, 1086, 0, 860, 0], [40316, 0, -6246, 0, 0, 0, 0, 0], [0, 114, 0, -114, 16510, 0, 228, 0], [0, 1275, 0, -1275, -14524, 0, 2550, 0], [0, -6393, 0, 0, 0, -166, 0, 0], [0, -2038, 0, -22464, 0, 0, 0, -4076], [-51776, 0, -3594, 0, 0, 0, 0, 0], [0, 2349, 0, -34841, 0, 0, 0, 4698], [0, -1410, 0, 1410, 15634, 0, -2820, 0], [101644, 0, 5666, 0, 0, 0, 0, 0], [0, -1911, 0, -7897, 0, 0, 0, -3822], [146824, 0, -10314, 0, 0, 0, 0, 0], [0, -3141, 0, -47252, 0, 0, 0, -6282], [-50394, 0, 9494, 0, 0, 0, 0, 0], [0, 6820, 0, 0, 0, -814, 0, 0], [0, 2851, 0, 0, 0, -426, 0, 0], [-111304, 0, -1086, 0, 0, 0, 0, 0], [102272, 0, 11890, 0, 0, 0, 0, 0], [0, -942, 0, 28684, 0, 0, 0, -1884], [0, -4506, 0, 0, 0, 932, 0, 0], [0, -1579, 0, 1579, 16580, 0, -3158, 0], [-259616, 0, -4254, 0, 0, 0, 0, 0], [0, -1810, 0, 1810, -13254, 0, -3620, 0], [0, 7681, 0, 0, 0, -202, 0, 0], [0, -1899, 0, 1899, 26380, 0, -3798, 0], [0, 3022, 0, 8251, 0, 0, 0, 6044], [0, 560, 0, 0, 0, 1608, 0, 0], [0, 605, 0, -605, 32146, 0, 1210, 0], [-25484, 0, -6066, 0, 0, 0, 0, 0], [133508, 0, -3510, 0, 0, 0, 0, 0], [0, -549, 0, -59713, 0, 0, 0, -1098], [0, -11035, 0, 0, 0, -914, 0, 0], [0, 574, 0, 0, 0, 3220, 0, 0], [0, 4130, 0, -14341, 0, 0, 0, 8260], [-31574, 0, -24496, 0, 0, 0, 0, 0], [0, -165, 0, 165, -19204, 0, -330, 0], [0, -1410, 0, 1410, 58266, 0, -2820, 0], [0, 4566, 0, 95525, 0, 0, 0, 9132], [-222056, 0, 4056, 0, 0, 0, 0, 0], [217436, 0, -26946, 0, 0, 0, 0, 0], [0, -2662, 0, -73432, 0, 0, 0, -5324], [0, 8648, 0, -41829, 0, 0, 0, 17296], [0, -11718, 0, -8032, 0, 0, 0, -23436], [236584, 0, -11184, 0, 0, 0, 0, 0], [0, 30112, 0, 0, 0, 534, 0, 0], [0, 10303, 0, 25617, 0, 0, 0, 20606], [0, 830, 0, -830, -86906, 0, 1660, 0], [0, -5430, 0, 5430, 5666, 0, -10860, 0], [0, -2905, 0, 0, 0, -1418, 0, 0], [-214064, 0, -12256, 0, 0, 0, 0, 0], [-16824, 0, 20324, 0, 0, 0, 0, 0], [0, -10608, 0, -18677, 0, 0, 0, -21216], [0, 3825, 0, 177737, 0, 0, 0, 7650], [0, 666, 0, -666, 101330, 0, 1332, 0], [-59492, 0, 10710, 0, 0, 0, 0, 0], [0, 17289, 0, 0, 0, -3956, 0, 0], [0, 6196, 0, -6196, 18880, 0, 12392, 0], [0, -7671, 0, -7348, 0, 0, 0, -15342], [0, 2385, 0, -2385, -12096, 0, 4770, 0], [0, 6717, 0, 0, 0, 818, 0, 0], [-58648, 0, 38770, 0, 0, 0, 0, 0], [-260442, 0, -37100, 0, 0, 0, 0, 0], [0, 31356, 0, 0, 0, -2376, 0, 0], [0, -6771, 0, -12983, 0, 0, 0, -13542], [0, -9647, 0, -18336, 0, 0, 0, -19294], [7004, 0, -22494, 0, 0, 0, 0, 0], [0, 651, 0, -651, -40810, 0, 1302, 0], [-95688, 0, 6790, 0, 0, 0, 0, 0], [0, -4030, 0, 4030, -148154, 0, -8060, 0], [0, 266, 0, 0, 0, -6932, 0, 0], [0, 35792, 0, 0, 0, 1536, 0, 0], [0, 5401, 0, 23068, 0, 0, 0, 10802], [0, 6720, 0, -14891, 0, 0, 0, 13440], [0, 5181, 0, -4, 0, 0, 0, 10362], [-92986, 0, -20414, 0, 0, 0, 0, 0], [0, -32369, 0, 0, 0, 1918, 0, 0], [238552, 0, -49650, 0, 0, 0, 0, 0], [0, -7585, 0, 7585, 19064, 0, -15170, 0], [0, 10175, 0, -10175, 9376, 0, 20350, 0], [0, 20847, 0, 0, 0, 2052, 0, 0], [0, 23734, 0, 0, 0, 2492, 0, 0], [0, -809, 0, 237492, 0, 0, 0, -1618], [0, -5280, 0, 85640, 0, 0, 0, -10560], [0, 8241, 0, 0, 0, -6422, 0, 0], [0, 5755, 0, -5755, 74804, 0, 11510, 0], [0, 554, 0, -554, 158030, 0, 1108, 0], [0, -60834, 0, 0, 0, 810, 0, 0], [0, 9885, 0, -9885, -11796, 0, 19770, 0], [0, 23687, 0, 75447, 0, 0, 0, 47374], [0, -1491, 0, 1491, 49100, 0, -2982, 0], [-384704, 0, 16734, 0, 0, 0, 0, 0], [-555976, 0, -23064, 0, 0, 0, 0, 0], [0, -18434, 0, 0, 0, -2950, 0, 0], [0, 1820, 0, -402076, 0, 0, 0, 3640], [0, -50535, 0, 0, 0, 854, 0, 0], [0, 7919, 0, -7919, -62560, 0, 15838, 0], [0, 660, 0, -660, -6276, 0, 1320, 0], [0, 3189, 0, -3189, 101410, 0, 6378, 0], [0, 5605, 0, 0, 0, 6202, 0, 0], [0, 3140, 0, -3140, -115104, 0, 6280, 0], [0, -28443, 0, 62105, 0, 0, 0, -56886], [1135336, 0, -9436, 0, 0, 0, 0, 0], [0, -12053, 0, 66300, 0, 0, 0, -24106]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_35_c_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_35_5_c_e();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. function MakeNewformModFrm_35_5_c_e(:prec:=8) chi := MakeCharacter_35_c(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 5)); S := BaseChange(S, Kf); maxprec := NextPrime(997) - 1; while true do trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]); B := Basis(S, prec + 1); S_basismat := Matrix([AbsEltseq(g): g in B]); if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then S_basismat := ChangeRing(S_basismat,Kf); f_lincom := Solution(S_basismat,trunc_vec); f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]]; return f; end if; error if prec eq maxprec, "Unable to distinguish newform within newspace"; prec := Min(Ceiling(1.25 * prec), maxprec); end while; end function; // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_35_5_c_e();". // This may take a long time! To see verbose output, uncomment the SetVerbose line below. // The default sign is -1. You can change this with the optional parameter "sign". function MakeNewformModSym_35_5_c_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_35_c(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,5,sign))); Vf := Kernel([<2,R![1194, 0, 75, 0, 1]>,<3,R![-90, 0, 1]>],Snew); return Vf; end function;