// Make newform 1296.3.q.e in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1296_q();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1296_q_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_1296_3_q_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_1296_3_q_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 := [4, 0, -2, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 4, 0, 0], [0, 0, 1, 0], [0, 0, 0, 2]]; Rf_basisdens := [1, 1, 2, 1]; 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_1296_q();" function MakeCharacter_1296_q() N := 1296; order := 6; char_gens := [1135, 325, 1217]; v := [6, 6, 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_1296_q_Hecke();" function MakeCharacter_1296_q_Hecke(Kf) N := 1296; order := 6; char_gens := [1135, 325, 1217]; char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [1, 0, -1, 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 := 3; raw_aps := [[0, 0, 0, 0], [0, 0, 0, 0], [0, 1, 0, 0], [0, 0, -6, 0], [0, 1, 0, -1], [-10, 0, 10, 0], [0, 0, 0, 4], [-2, 0, 0, 0], [0, 2, 0, 0], [0, -3, 0, 3], [-22, 0, 22, 0], [-6, 0, 0, 0], [0, 6, 0, 0], [0, 0, 82, 0], [0, 12, 0, -12], [0, 0, 0, 11], [0, -13, 0, 0], [0, 0, 86, 0], [2, 0, -2, 0], [0, 0, 0, -22], [82, 0, 0, 0], [0, 0, 10, 0], [0, -13, 0, 13], [0, 0, 0, 6], [0, 0, 94, 0], [0, -9, 0, 9], [-134, 0, 134, 0], [0, 0, 0, 9], [10, 0, 0, 0], [0, -12, 0, 0], [-106, 0, 0, 0], [0, 1, 0, 0], [0, 18, 0, -18], [-78, 0, 78, 0], [0, 29, 0, 0], [0, 0, 218, 0], [86, 0, -86, 0], [222, 0, 0, 0], [0, 30, 0, 0], [0, 33, 0, -33], [0, 0, 0, 27], [90, 0, 0, 0], [0, -48, 0, 48], [-2, 0, 2, 0], [0, 0, 0, 15], [-250, 0, 0, 0], [34, 0, -34, 0], [0, 0, -278, 0], [0, 39, 0, -39], [-58, 0, 58, 0], [0, 0, 0, -70], [0, 4, 0, 0], [0, 0, 30, 0], [0, 0, 0, -19], [0, 32, 0, 0], [0, -38, 0, 38], [0, 0, 0, -71], [-266, 0, 0, 0], [0, 0, -346, 0], [0, 22, 0, -22], [-46, 0, 46, 0], [0, -39, 0, 0], [30, 0, 0, 0], [0, -102, 0, 0], [0, 0, -210, 0], [0, -27, 0, 27], [0, 0, 434, 0], [510, 0, -510, 0], [0, 27, 0, 0], [0, 0, -426, 0], [0, -8, 0, 8], [0, 0, 0, -78], [0, 0, -566, 0], [-218, 0, 218, 0], [142, 0, 0, 0], [0, 0, 0, 0], [0, -97, 0, 97], [-310, 0, 0, 0], [0, 60, 0, 0], [270, 0, -270, 0], [0, 9, 0, 0], [0, 0, 454, 0], [0, 0, 0, 44], [706, 0, 0, 0], [0, 0, -486, 0], [0, 125, 0, -125], [0, 0, 0, 128], [0, 0, -338, 0], [0, 137, 0, -137], [74, 0, -74, 0], [0, 0, 0, -141], [0, 56, 0, -56], [134, 0, 0, 0], [0, -9, 0, 0], [-30, 0, 30, 0], [0, 0, 0, -42], [0, -21, 0, 0], [0, 0, 0, -14], [494, 0, 0, 0], [234, 0, 0, 0], [0, 0, 290, 0], [0, 0, 0, -31], [0, -43, 0, 0], [0, 110, 0, -110], [402, 0, -402, 0], [98, 0, 0, 0], [0, 97, 0, -97], [0, 0, 0, -124], [0, 114, 0, 0], [0, 0, 398, 0], [170, 0, -170, 0], [-1030, 0, 0, 0], [0, -186, 0, 0], [0, 0, -14, 0], [-1114, 0, 0, 0], [0, 80, 0, -80], [-798, 0, 798, 0], [0, 0, 0, 26], [0, -57, 0, 0], [0, -141, 0, 141], [-986, 0, 986, 0], [0, 0, -34, 0], [0, 71, 0, -71], [0, 0, 0, -23], [0, 0, 578, 0], [0, 0, 0, -219], [0, 0, 1222, 0], [0, 0, 0, -44], [0, 0, -870, 0], [214, 0, -214, 0], [958, 0, 0, 0], [0, -178, 0, 0], [-630, 0, 630, 0], [602, 0, 0, 0], [0, 194, 0, 0], [-770, 0, 770, 0], [0, 0, 0, -33], [514, 0, -514, 0], [0, -125, 0, 0], [0, 0, 0, -134], [1454, 0, 0, 0], [0, 171, 0, -171], [-166, 0, 166, 0], [0, 0, 0, 173], [1258, 0, 0, 0], [0, 234, 0, -234], [0, 0, 742, 0], [0, -42, 0, 42], [-1230, 0, 1230, 0], [0, 0, 0, -8], [822, 0, -822, 0], [0, 0, 0, -116], [-962, 0, 0, 0], [0, 202, 0, 0], [0, 0, 1042, 0], [0, -284, 0, 284], [614, 0, 0, 0], [0, -8, 0, 8], [-462, 0, 0, 0], [0, 63, 0, 0], [0, 123, 0, -123], [0, 0, 0, 270], [-70, 0, 70, 0], [0, 0, 0, -111], [0, -180, 0, 0], [0, -114, 0, 114], [854, 0, 0, 0], [0, 0, 518, 0], [-958, 0, 0, 0], [0, 37, 0, 0], [0, 285, 0, -285], [-426, 0, 426, 0], [0, -314, 0, 0], [0, 0, 1166, 0], [-694, 0, 694, 0], [0, 74, 0, 0], [0, 0, -1582, 0], [0, 0, 0, -153], [-826, 0, 0, 0], [0, 0, 886, 0], [0, 0, -598, 0], [0, 95, 0, -95], [-1338, 0, 1338, 0], [0, 0, 0, 114], [0, -84, 0, 0], [0, -141, 0, 141], [1258, 0, 0, 0], [0, 0, 450, 0], [-1714, 0, 1714, 0], [0, 0, 0, -96], [130, 0, 0, 0], [0, 81, 0, -81], [-34, 0, 0, 0], [0, 221, 0, -221], [0, 0, 0, -249], [0, -298, 0, 0], [1598, 0, -1598, 0], [0, 0, -298, 0], [0, -272, 0, 272], [0, 0, 0, -54], [0, -73, 0, 0], [0, 0, 1034, 0], [-1562, 0, 1562, 0], [0, 0, -1186, 0], [0, 0, 0, 153], [0, 0, 0, 309], [-1706, 0, 0, 0], [0, 369, 0, 0], [0, -446, 0, 446], [-494, 0, 494, 0], [1698, 0, 0, 0], [0, -3, 0, 0], [0, 0, 730, 0], [0, 197, 0, -197], [0, 94, 0, 0], [0, 0, -1298, 0], [1610, 0, -1610, 0], [0, 404, 0, -404], [0, 0, 0, -306], [0, 339, 0, 0], [-890, 0, 890, 0], [1818, 0, -1818, 0], [0, 336, 0, 0], [310, 0, 0, 0], [0, 477, 0, 0], [0, 0, -794, 0], [0, 150, 0, -150], [0, 0, 0, 360], [0, 0, -390, 0], [0, -263, 0, 263], [1014, 0, -1014, 0], [-1090, 0, 0, 0], [1898, 0, -1898, 0], [0, -242, 0, 0], [0, 0, -1902, 0], [0, 212, 0, -212], [766, 0, -766, 0], [0, 0, 0, -397], [0, 283, 0, 0], [0, 0, 0, 178], [0, -69, 0, 69], [-2450, 0, 0, 0], [-326, 0, 326, 0], [-374, 0, 0, 0], [0, -468, 0, 0], [0, -34, 0, 34], [2006, 0, 0, 0], [0, -327, 0, 0], [-2702, 0, 2702, 0], [0, 0, 0, 44], [2262, 0, -2262, 0], [0, 0, 0, -240], [0, 502, 0, 0], [0, 0, -1138, 0], [0, -279, 0, 279], [0, 0, 0, -93], [1370, 0, 0, 0], [0, 0, 466, 0], [0, 0, 0, -489], [178, 0, 0, 0], [0, 0, 2602, 0], [0, -225, 0, 225], [-2170, 0, 2170, 0], [-534, 0, 0, 0], [0, 40, 0, 0], [0, 0, -958, 0], [0, 0, 0, 97], [0, 94, 0, -94], [-1294, 0, 1294, 0], [0, -79, 0, 0], [2870, 0, -2870, 0], [-2594, 0, 0, 0], [0, 0, 494, 0], [-1238, 0, 1238, 0], [2846, 0, -2846, 0], [1126, 0, 0, 0], [0, -141, 0, 0], [0, 0, 3094, 0], [-302, 0, 0, 0], [0, -77, 0, 77], [0, 440, 0, 0], [442, 0, -442, 0], [0, 454, 0, -454], [0, 0, 1606, 0], [3218, 0, -3218, 0], [0, 0, 0, -140], [2210, 0, 0, 0], [0, -243, 0, 0], [0, 0, 2906, 0], [0, 0, 0, 88], [0, 113, 0, -113], [0, 0, 0, 171], [0, 370, 0, 0], [0, 591, 0, 0], [0, 0, -3210, 0], [0, -125, 0, 0], [0, 0, -3670, 0], [0, -597, 0, 597], [0, 0, 0, 189], [0, 0, -670, 0], [-2162, 0, 2162, 0], [0, 0, 0, 361], [-458, 0, 0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1296_q_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1296_3_q_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_1296_3_q_e(:prec:=4) chi := MakeCharacter_1296_q(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 3)); S := BaseChange(S, Kf); maxprec := NextPrime(1999) - 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_1296_3_q_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_1296_3_q_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1296_q(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign))); Vf := Kernel([<5,R![1024, 0, -32, 0, 1]>,<7,R![36, 6, 1]>],Snew); return Vf; end function;