// Make newform 1024.2.g.e in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1024_g();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1024_g_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_1024_2_g_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_1024_2_g_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 := [6561, 0, -5832, 0, 2592, 0, -576, 0, 127, 0, -64, 0, 32, 0, -8, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1533816, 0, -708993, 0, 152253, 0, -4064, 0, 13829, 0, -8260, 0, 1975, 0, -23], [0, -9086985, 0, 3232224, 0, -946332, 0, 22850, 0, -82925, 0, 44518, 0, -13663, 0, 683], [654642, 0, 206145, 0, -270972, 0, 70408, 0, 5156, 0, 1895, 0, -2582, 0, 928, 0], [0, -2899962, 0, 748116, 0, 339264, 0, -36395, 0, -29215, 0, 12497, 0, 1630, 0, -998], [-72738, 0, 119160, 0, -43364, 0, 10513, 0, -828, 0, 1352, 0, -606, 0, 152, 0], [-10136016, 0, 6035391, 0, -1456047, 0, 123065, 0, -104255, 0, 69988, 0, -19279, 0, 1529, 0], [1458000, 0, -998244, 0, 308016, 0, -29444, 0, 13856, 0, -11716, 0, 4297, 0, -536, 0], [2030022, 0, -331344, 0, -19293, 0, 27322, 0, 23963, 0, -5671, 0, 445, 0, 541, 0], [0, 2404728, 0, -1523259, 0, 412023, 0, -35612, 0, 17417, 0, -15376, 0, 5749, 0, -545], [39366, 0, -283905, 0, 121995, 0, -22760, 0, 53, 0, -2110, 0, 1360, 0, -269, 0], [0, -2566080, 0, 3232629, 0, -1021203, 0, 159412, 0, -14089, 0, 39080, 0, -12281, 0, 2011], [-1452897, 0, 2014875, 0, -763695, 0, 116027, 0, -13121, 0, 15790, 0, -8245, 0, 2069, 0], [0, -12535155, 0, 7802001, 0, -2182401, 0, 129686, 0, -135176, 0, 90646, 0, -30910, 0, 2318], [0, -16315020, 0, 9057663, 0, -2289825, 0, 168104, 0, -158573, 0, 106156, 0, -30895, 0, 3323], [0, 11764602, 0, -10649475, 0, 3746691, 0, -632545, 0, 112300, 0, -118331, 0, 48179, 0, -9367]]; Rf_basisdens := [1, 269001, 1883007, 627669, 1883007, 69741, 627669, 209223, 209223, 627669, 89667, 1883007, 627669, 1883007, 1883007, 1883007]; 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_1024_g();" function MakeCharacter_1024_g() N := 1024; order := 8; char_gens := [1023, 5]; v := [8, 7]; // 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_1024_g_Hecke();" function MakeCharacter_1024_g_Hecke(Kf) N := 1024; order := 8; char_gens := [1023, 5]; char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, 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 := 2; raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0], [0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [-1, 0, 0, -2, 0, 1, -1, -1, -1, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, -2, 0, 1, 1, 0, 0, 0, -1, 0, 0, 0], [0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, -2, 0, -1, -1, 1], [0, -4, -1, 0, 0, 0, 0, 0, 0, 2, 0, -3, 0, 0, -2, -1], [0, 0, 0, -2, 0, 1, 0, 1, 1, 0, -1, 0, -1, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0], [3, 0, 0, -1, 0, -4, -1, 4, 0, 0, 1, 0, -1, 0, 0, 0], [1, 0, 0, 0, 0, 2, -2, 3, -2, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, -1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0], [0, -2, -1, 0, 1, 0, 0, 0, 0, 2, 0, -2, 0, -1, 0, 1], [4, 0, 0, 4, 0, -4, 0, -3, -1, 0, 0, 0, 0, 0, 0, 0], [0, -4, 1, 0, 0, 0, 0, 0, 0, 3, 0, -1, 0, -1, -2, -1], [-2, 0, 0, 1, 0, -3, 0, 0, -2, 0, 2, 0, 3, 0, 0, 0], [0, 3, -1, 0, 2, 0, 0, 0, 0, -1, 0, 1, 0, 0, 1, 2], [0, 0, 2, 0, -1, 0, 0, 0, 0, 2, 0, 0, 0, -1, -3, -2], [-1, 0, 0, 1, 0, 0, 0, 3, 0, 0, 1, 0, -1, 0, 0, 0], [0, -6, -3, 0, 3, 0, 0, 0, 0, 6, 0, -3, 0, 1, -3, -1], [0, -2, 0, 0, 1, 0, 0, 0, 0, 4, 0, 0, 0, 1, -2, -1], [2, 0, 0, 0, 0, 0, 1, 2, -1, 0, -2, 0, -2, 0, 0, 0], [2, 0, 0, 2, 0, 0, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0], [-5, 0, 0, 5, 0, -2, 1, 2, 0, 0, 1, 0, 1, 0, 0, 0], [0, 0, 2, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 3, 3, -2], [0, 5, 2, 0, 1, 0, 0, 0, 0, -2, 0, 7, 0, 2, 0, 0], [5, 0, 0, -1, 0, -5, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, -1, 0, 0, -1, -5, 1, 0, 1, 0, -1, 0, 0, 0], [0, -4, 2, 0, 2, 0, 0, 0, 0, 0, 0, -1, 0, 2, -1, 2], [0, 5, -1, 0, -2, 0, 0, 0, 0, -2, 0, 2, 0, 0, 1, -2], [-2, 0, 0, 1, 0, 6, 3, -2, 3, 0, 1, 0, -1, 0, 0, 0], [0, 0, 2, 0, 3, 0, 0, 0, 0, 4, 0, -4, 0, 2, 0, 0], [-5, 0, 0, -6, 0, -8, -1, -7, -1, 0, 0, 0, 1, 0, 0, 0], [0, 4, -1, 0, 4, 0, 0, 0, 0, -2, 0, -1, 0, -4, 2, -1], [-4, 0, 0, 3, 0, -1, 0, -6, 2, 0, -2, 0, -1, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 2, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 1, -2, 0], [3, 0, 0, 6, 0, -3, -3, 9, -3, 0, -3, 0, 0, 0, 0, 0], [0, 6, 0, 0, -3, 0, 0, 0, 0, -3, 0, 9, 0, -3, 6, 3], [3, 0, 0, 0, 0, 4, -3, 7, -3, 0, 0, 0, 3, 0, 0, 0], [0, -6, -6, 0, -6, 0, 0, 0, 0, 0, 0, -5, 0, -2, -5, -2], [-2, 0, 0, 2, 0, 4, 0, -1, 1, 0, 0, 0, 1, 0, 0, 0], [2, 0, 0, -1, 0, -4, -4, 4, 0, 0, -3, 0, -4, 0, 0, 0], [0, 0, 2, 0, -1, 0, 0, 0, 0, -9, 0, 0, 0, -1, -2, -2], [0, -4, 0, 0, -1, 0, 0, 0, 0, 8, 0, 0, 0, 5, -4, 1], [0, 0, -4, 0, -4, 0, 0, 0, 0, 0, 0, 3, 0, 0, 3, 0], [0, 0, 1, 0, 4, 0, 0, 0, 0, -2, 0, 2, 0, 0, -4, 4], [-6, 0, 0, 5, 0, 2, 0, -2, 0, 0, -1, 0, 0, 0, 0, 0], [-3, 0, 0, 1, 0, 4, 2, 1, 2, 0, 1, 0, -1, 0, 0, 0], [0, 6, -1, 0, 1, 0, 0, 0, 0, -6, 0, 5, 0, 3, 1, -3], [4, 0, 0, -2, 0, -2, -4, 11, -1, 0, 4, 0, 1, 0, 0, 0], [0, -6, 3, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, -3, 0, 3], [-15, 0, 0, -5, 0, -4, 1, -1, 1, 0, 1, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, -11, 0], [1, 0, 0, 1, 0, -1, 1, 0, 4, 0, 4, 0, 0, 0, 0, 0], [0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, -2, 0, 5, 2, -5], [0, 0, 0, 4, 0, 2, 4, 1, 1, 0, 0, 0, -4, 0, 0, 0], [7, 0, 0, -1, 0, 0, 0, 7, 0, 0, 5, 0, 5, 0, 0, 0], [0, 11, 5, 0, 0, 0, 0, 0, 0, -7, 0, 4, 0, -5, 8, 1], [8, 0, 0, -5, 0, 8, 2, -8, 0, 0, 5, 0, 2, 0, 0, 0], [0, -1, 0, 0, 5, 0, 0, 0, 0, -1, 0, -3, 0, -1, -1, -5], [0, 2, 5, 0, 6, 0, 0, 0, 0, -1, 0, -3, 0, -6, 1, 5], [-7, 0, 0, 8, 0, 0, 2, -7, -2, 0, -2, 0, -2, 0, 0, 0], [8, 0, 0, -4, 0, -1, 0, 13, -1, 0, 1, 0, -7, 0, 0, 0], [0, 3, 4, 0, 5, 0, 0, 0, 0, 5, 0, -2, 0, 4, 0, 0], [0, 0, 0, -2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -10, 1, 0, 0, 0, 0, 0, 0, 6, 0, -4, 0, -1, -8, -1], [2, 0, 0, -4, 0, -3, 0, 9, -3, 0, 3, 0, 1, 0, 0, 0], [1, 0, 0, -15, 0, -18, 1, 1, -1, 0, 1, 0, -1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, -5, 0], [0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, -4, 0, -1, 4, 1], [0, 0, 0, -2, 0, -4, -2, -1, -3, 0, 0, 0, 2, 0, 0, 0], [0, -4, -3, 0, 0, 0, 0, 0, 0, 2, 0, -2, 0, 3, -4, -9], [0, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, -2, 2, -2], [-3, 0, 0, 7, 0, 0, -3, 0, 0, 0, 1, 0, -3, 0, 0, 0], [-5, 0, 0, -17, 0, 5, 1, -18, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 16, 0, -14, 2, -1, -1, 0, -2, 0, 1, 0, 0, 0], [-9, 0, 0, 4, 0, 0, 4, -9, -4, 0, 2, 0, 2, 0, 0, 0], [0, -11, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -11, 0], [-6, 0, 0, 1, 0, 10, 6, -10, 0, 0, 1, 0, 6, 0, 0, 0], [0, -8, 3, 0, -3, 0, 0, 0, 0, 8, 0, -7, 0, -1, -1, 1], [-2, 0, 0, -8, 0, 10, 2, 3, 3, 0, -2, 0, -3, 0, 0, 0], [0, -2, 3, 0, 0, 0, 0, 0, 0, 1, 0, -5, 0, 0, -1, 3], [0, -17, -7, 0, 0, 0, 0, 0, 0, 12, 0, -5, 0, 7, -10, 1], [-6, 0, 0, 2, 0, 4, 8, -5, 5, 0, 8, 0, 5, 0, 0, 0], [3, 0, 0, 0, 0, 14, 4, -11, 4, 0, 0, 0, 0, 0, 0, 0], [-9, 0, 0, 3, 0, 9, 3, 0, -6, 0, -6, 0, 0, 0, 0, 0], [0, -8, 3, 0, -3, 0, 0, 0, 0, 8, 0, -5, 0, -1, -3, 1], [0, -1, 0, 0, -3, 0, 0, 0, 0, -2, 0, -4, 0, -13, -1, 3], [0, -4, 2, 0, 2, 0, 0, 0, 0, 0, 0, -9, 0, 2, -9, 2], [0, 0, -2, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, -1, 6, 2], [0, 2, -6, 0, 7, 0, 0, 0, 0, 3, 0, -1, 0, -6, 0, 0], [0, -10, 0, 0, 7, 0, 0, 0, 0, 9, 0, -11, 0, 1, -10, -7], [0, -16, -1, 0, 0, 0, 0, 0, 0, 8, 0, -3, 0, 0, -8, -1], [16, 0, 0, 7, 0, 13, 0, -4, -2, 0, 2, 0, 1, 0, 0, 0], [8, 0, 0, 1, 0, 26, 1, -8, 1, 0, 1, 0, -1, 0, 0, 0], [0, -5, 2, 0, 5, 0, 0, 0, 0, -4, 0, -1, 0, 2, 0, 0], [2, 0, 0, 15, 0, 1, 0, -14, 0, 0, 0, 0, -1, 0, 0, 0], [0, 5, 11, 0, 2, 0, 0, 0, 0, 1, 0, -1, 0, 0, 7, 2], [-19, 0, 0, -2, 0, 19, 1, -3, 3, 0, 3, 0, 0, 0, 0, 0], [0, 13, 0, 0, -1, 0, 0, 0, 0, -12, 0, 14, 0, -5, 13, 1], [-8, 0, 0, -1, 0, 0, -3, -8, 3, 0, 5, 0, 5, 0, 0, 0], [0, -5, 5, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, -5, -10, 1], [-10, 0, 0, 2, 0, 2, 8, -4, 4, 0, 8, 0, 4, 0, 0, 0], [0, 5, 10, 0, 1, 0, 0, 0, 0, -1, 0, 6, 0, 10, 0, 0], [2, 0, 0, -6, 0, 4, -2, -12, 0, 0, 2, 0, 0, 0, 0, 0], [0, 10, -1, 0, -4, 0, 0, 0, 0, -5, 0, 5, 0, 4, 5, -1], [-6, 0, 0, -20, 0, 0, -3, -6, 3, 0, -2, 0, -2, 0, 0, 0], [0, -2, 4, 0, 4, 0, 0, 0, 0, 0, 0, 2, 0, -4, 2, -4], [0, 0, 0, 1, 0, -2, 4, 2, 0, 0, 5, 0, 4, 0, 0, 0], [-6, 0, 0, 2, 0, 8, 1, 8, 1, 0, 2, 0, -2, 0, 0, 0], [0, -7, -4, 0, -7, 0, 0, 0, 0, -2, 0, -5, 0, -4, 0, 0], [0, 4, -3, 0, 2, 0, 0, 0, 0, -2, 0, -13, 0, -2, 2, -3], [-2, 0, 0, 14, 0, 4, 0, 5, -5, 0, 0, 0, -5, 0, 0, 0], [0, -17, -1, 0, 2, 0, 0, 0, 0, 9, 0, -9, 0, 0, 1, 2], [0, 0, 2, 0, -5, 0, 0, 0, 0, 6, 0, 0, 0, -5, -1, -2], [-3, 0, 0, 1, 0, 3, 1, 0, -4, 0, -4, 0, 0, 0, 0, 0], [0, -8, 0, 0, 9, 0, 0, 0, 0, 8, 0, -8, 0, 13, -8, -9], [-1, 0, 0, 0, 0, 4, 1, -5, 9, 0, 0, 0, -1, 0, 0, 0], [14, 0, 0, 10, 0, 4, 8, -1, 1, 0, 8, 0, 1, 0, 0, 0], [-7, 0, 0, 11, 0, -4, 5, 4, 0, 0, 9, 0, 5, 0, 0, 0], [0, -4, -8, 0, -5, 0, 0, 0, 0, 2, 0, -6, 0, -8, 0, 0], [0, -6, 0, 0, -1, 0, 0, 0, 0, 6, 0, -6, 0, -3, -6, 1], [-6, 0, 0, -21, 0, 3, 0, 12, 0, 0, 0, 0, 9, 0, 0, 0], [5, 0, 0, 1, 0, -14, 1, 14, 0, 0, 7, 0, 1, 0, 0, 0], [0, 20, -1, 0, 1, 0, 0, 0, 0, -20, 0, 9, 0, -5, 11, 5], [0, 26, 7, 0, 0, 0, 0, 0, 0, -13, 0, 15, 0, 0, 13, 7], [-12, 0, 0, -10, 0, -5, 0, 3, 1, 0, -1, 0, 5, 0, 0, 0], [0, 18, 1, 0, 10, 0, 0, 0, 0, -5, 0, 5, 0, 0, 8, 10], [0, 0, 0, 0, -11, 0, 0, 0, 0, -7, 0, 0, 0, -11, -8, 0], [0, 8, -3, 0, 3, 0, 0, 0, 0, -8, 0, 3, 0, 1, 5, -1], [-19, 0, 0, -18, 0, 4, 1, -3, 7, 0, 0, 0, -1, 0, 0, 0], [5, 0, 0, 10, 0, 0, 6, 5, -6, 0, 4, 0, 4, 0, 0, 0], [-4, 0, 0, 4, 0, -4, 2, 3, -3, 0, 2, 0, -3, 0, 0, 0], [13, 0, 0, -19, 0, 10, 1, -10, 0, 0, -5, 0, 1, 0, 0, 0], [0, 8, 0, 0, -3, 0, 0, 0, 0, -7, 0, 9, 0, 5, 8, 3], [-4, 0, 0, -1, 0, -7, 0, 4, 2, 0, -2, 0, -7, 0, 0, 0], [7, 0, 0, 0, 0, -22, -8, 9, -8, 0, 0, 0, 0, 0, 0, 0], [0, 5, 2, 0, 1, 0, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0], [4, 0, 0, -2, 0, -4, -6, 3, -7, 0, 0, 0, 6, 0, 0, 0], [0, 18, -1, 0, 0, 0, 0, 0, 0, -9, 0, 21, 0, 0, 9, -1], [0, -13, 1, 0, 0, 0, 0, 0, 0, 9, 0, -4, 0, -1, -8, -7], [4, 0, 0, 23, 0, 1, 0, -20, -2, 0, 2, 0, 1, 0, 0, 0], [0, 0, -10, 0, -5, 0, 0, 0, 0, -3, 0, 0, 0, -5, 2, 10], [-12, 0, 0, -12, 0, 8, 0, 5, 3, 0, 0, 0, 0, 0, 0, 0], [-12, 0, 0, 9, 0, 0, -3, -12, 3, 0, 3, 0, 3, 0, 0, 0], [0, 10, 5, 0, 0, 0, 0, 0, 0, -4, 0, 6, 0, -5, 12, 11], [0, -18, 6, 0, 6, 0, 0, 0, 0, 0, 0, -9, 0, -6, -9, -6], [23, 0, 0, -1, 0, -23, 3, -4, -2, 0, -2, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 2, 2, 0, 0, 0, -2, 0, 0, 0, 0, 0], [0, 11, 0, 0, -15, 0, 0, 0, 0, -12, 0, 10, 0, -1, 11, 15], [0, -18, 1, 0, 2, 0, 0, 0, 0, 9, 0, -13, 0, -2, -9, 1], [0, 16, 0, 0, -1, 0, 0, 0, 0, 8, 0, 8, 0, 0, 0, 0], [0, 26, 1, 0, -1, 0, 0, 0, 0, -26, 0, 14, 0, 1, 12, -1], [0, -16, 3, 0, 0, 0, 0, 0, 0, 8, 0, -9, 0, 0, -8, 3], [-2, 0, 0, -2, 0, -4, -4, 3, -3, 0, -4, 0, -3, 0, 0, 0], [7, 0, 0, -1, 0, 12, 8, -25, 8, 0, -1, 0, 1, 0, 0, 0], [5, 0, 0, -19, 0, -5, -1, -18, 2, 0, 2, 0, 0, 0, 0, 0], [0, -3, 0, 0, -3, 0, 0, 0, 0, 9, 0, 3, 0, 3, -3, 3], [22, 0, 0, 21, 0, 0, -1, 22, 1, 0, -1, 0, -1, 0, 0, 0], [0, 0, -6, 0, 7, 0, 0, 0, 0, 7, 0, 0, 0, 7, 2, 6], [0, 6, -6, 0, 3, 0, 0, 0, 0, -3, 0, 9, 0, -6, 0, 0], [2, 0, 0, 0, 0, -2, -2, -3, 9, 0, 2, 0, -9, 0, 0, 0], [0, -10, -11, 0, -2, 0, 0, 0, 0, 5, 0, 1, 0, 2, -5, -11], [0, 4, 10, 0, 10, 0, 0, 0, 0, 0, 0, -4, 0, 2, -4, 2], [-13, 0, 0, 5, 0, -6, 3, 6, 0, 0, -5, 0, 3, 0, 0, 0], [8, 0, 0, 6, 0, -14, -8, 5, -7, 0, 8, 0, 7, 0, 0, 0], [-15, 0, 0, -24, 0, -6, -9, 3, -9, 0, 0, 0, 9, 0, 0, 0], [0, -15, -3, 0, 0, 0, 0, 0, 0, 12, 0, -3, 0, 3, -6, -3], [10, 0, 0, -10, 0, 3, 0, 17, -1, 0, 1, 0, -5, 0, 0, 0], [0, 0, 6, 0, 7, 0, 0, 0, 0, 2, 0, 0, 0, 7, 1, -6], [-24, 0, 0, 0, 0, -12, 3, 18, 3, 0, 0, 0, 0, 0, 0, 0], [0, 8, -1, 0, 1, 0, 0, 0, 0, -8, 0, -2, 0, -5, 10, 5], [-5, 0, 0, -11, 0, 0, 4, -5, -4, 0, -1, 0, -1, 0, 0, 0], [0, -18, 3, 0, 0, 0, 0, 0, 0, 12, 0, -6, 0, -3, -12, -1], [-9, 0, 0, 1, 0, 6, -3, -6, 0, 0, -11, 0, -3, 0, 0, 0], [0, 0, 0, 0, 13, 0, 0, 0, 0, 1, 0, 0, 0, 13, -12, 0], [-3, 0, 0, 4, 0, 3, -5, 9, -9, 0, -9, 0, 0, 0, 0, 0], [0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, -4, 0, 2, -4, 2], [0, 16, 5, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 12], [10, 0, 0, -13, 0, 12, -6, -12, 0, 0, -9, 0, -6, 0, 0, 0], [2, 0, 0, -6, 0, -8, -7, 0, -7, 0, -6, 0, 6, 0, 0, 0], [0, 6, 9, 0, -9, 0, 0, 0, 0, -6, 0, 12, 0, -3, -6, 3], [-11, 0, 0, -18, 0, 16, -7, 17, -1, 0, 0, 0, 7, 0, 0, 0], [8, 0, 0, 3, 0, 11, 0, -6, -6, 0, 6, 0, 15, 0, 0, 0], [0, -2, 1, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -6], [-7, 0, 0, 4, 0, -2, -2, 19, -2, 0, 4, 0, -4, 0, 0, 0], [0, -14, -2, 0, -2, 0, 0, 0, 0, 0, 0, -4, 0, -6, -4, -6], [23, 0, 0, -13, 0, -14, -5, 3, -3, 0, -5, 0, -3, 0, 0, 0], [0, -16, -4, 0, 7, 0, 0, 0, 0, 1, 0, -17, 0, -4, 0, 0], [0, 9, 0, 0, 3, 0, 0, 0, 0, -15, 0, 3, 0, -13, 9, -3], [4, 0, 0, 6, 0, -3, 0, 1, -3, 0, 3, 0, -1, 0, 0, 0], [0, -12, -19, 0, -4, 0, 0, 0, 0, 5, 0, -5, 0, 0, -2, -4], [8, 0, 0, 4, 0, -4, 7, -14, 7, 0, 4, 0, -4, 0, 0, 0], [-4, 0, 0, -4, 0, 8, 4, -22, 2, 0, -4, 0, -2, 0, 0, 0], [-8, 0, 0, 7, 0, -13, 0, -2, -2, 0, 2, 0, -1, 0, 0, 0], [16, 0, 0, -14, 0, -4, -10, 0, 0, 0, -10, 0, 0, 0, 0, 0], [0, 0, -6, 0, -1, 0, 0, 0, 0, 4, 0, 0, 0, -1, 5, 6], [-11, 0, 0, 6, 0, 11, 9, -3, 11, 0, 11, 0, 0, 0, 0, 0], [0, 8, 3, 0, -3, 0, 0, 0, 0, -8, 0, -5, 0, -13, 13, 13], [4, 0, 0, 2, 0, -16, -2, -9, -7, 0, 0, 0, 2, 0, 0, 0], [-9, 0, 0, -3, 0, -8, -5, 5, -5, 0, -5, 0, -5, 0, 0, 0], [0, 3, 0, 0, -15, 0, 0, 0, 0, 9, 0, -6, 0, 0, 0, 0], [6, 0, 0, 26, 0, 11, 0, -31, -1, 0, 1, 0, 7, 0, 0, 0], [0, -20, -10, 0, -10, 0, 0, 0, 0, 0, 0, -22, 0, -6, -22, -6], [0, -12, -11, 0, -2, 0, 0, 0, 0, 10, 0, -10, 0, 0, 8, -2], [-11, 0, 0, 1, 0, -24, -6, 25, -6, 0, 1, 0, -1, 0, 0, 0], [0, 13, 6, 0, -7, 0, 0, 0, 0, 1, 0, 12, 0, 6, 0, 0], [0, 0, 0, -14, 0, 14, 0, 31, -1, 0, 0, 0, 1, 0, 0, 0], [11, 0, 0, 12, 0, -4, 1, -5, 1, 0, 0, 0, -1, 0, 0, 0], [0, -14, -3, 0, -6, 0, 0, 0, 0, 7, 0, 7, 0, 6, -7, -3], [0, -1, 3, 0, 0, 0, 0, 0, 0, -5, 0, -6, 0, -3, -12, 1], [0, 0, 2, 0, -5, 0, 0, 0, 0, -15, 0, 0, 0, -5, 14, -2], [-8, 0, 0, -1, 0, 26, 1, 4, 1, 0, -1, 0, 1, 0, 0, 0], [0, -2, -3, 0, 3, 0, 0, 0, 0, 2, 0, -1, 0, 13, -1, -13], [0, 0, 0, -6, 0, 6, 0, 45, -3, 0, 0, 0, 3, 0, 0, 0], [0, 10, 11, 0, 8, 0, 0, 0, 0, -5, 0, -13, 0, -8, 5, 11], [-12, 0, 0, -4, 0, -13, 0, 5, -1, 0, 1, 0, 1, 0, 0, 0], [-26, 0, 0, 25, 0, -18, 0, 18, 0, 0, -1, 0, 0, 0, 0, 0], [0, 28, 1, 0, -2, 0, 0, 0, 0, -14, 0, 21, 0, 2, 14, 1], [-2, 0, 0, 18, 0, 24, 4, -5, 5, 0, 4, 0, 5, 0, 0, 0], [0, -24, -3, 0, 3, 0, 0, 0, 0, 24, 0, -19, 0, -3, -5, 3], [0, -15, 0, 0, 1, 0, 0, 0, 0, 5, 0, -25, 0, 3, -15, -1], [24, 0, 0, 14, 0, -12, -10, -9, -3, 0, 0, 0, 10, 0, 0, 0], [-2, 0, 0, -2, 0, 0, -11, -2, 11, 0, -4, 0, -4, 0, 0, 0], [0, 12, -6, 0, -6, 0, 0, 0, 0, 0, 0, 12, 0, 6, 12, 6], [0, 0, 2, 0, 11, 0, 0, 0, 0, 4, 0, 0, 0, 11, -5, -2], [0, -4, -4, 0, -17, 0, 0, 0, 0, 1, 0, -5, 0, -4, 0, 0], [5, 0, 0, 3, 0, -5, -15, 18, -6, 0, -6, 0, 0, 0, 0, 0], [0, 17, 0, 0, -5, 0, 0, 0, 0, -25, 0, 9, 0, 11, 17, 5], [0, 22, -8, 0, -8, 0, 0, 0, 0, 0, 0, 8, 0, 8, 8, 8], [4, 0, 0, 6, 0, -16, 7, -6, 7, 0, 6, 0, -6, 0, 0, 0], [0, -22, -8, 0, -5, 0, 0, 0, 0, -2, 0, -20, 0, -8, 0, 0], [0, -20, 1, 0, -1, 0, 0, 0, 0, 20, 0, -3, 0, 5, -17, -5], [0, 0, 0, 34, 0, -34, 0, -11, 1, 0, 0, 0, -1, 0, 0, 0], [7, 0, 0, 8, 0, 18, 1, 19, -1, 0, 0, 0, -1, 0, 0, 0], [0, 29, 11, 0, 0, 0, 0, 0, 0, -22, 0, 7, 0, -11, 14, -3], [0, 0, 10, 0, 15, 0, 0, 0, 0, 11, 0, 0, 0, 15, 14, -10], [0, 6, 0, 0, -1, 0, 0, 0, 0, -5, 0, 7, 0, -3, 6, 1], [0, 21, -5, 0, 0, 0, 0, 0, 0, -16, 0, 5, 0, 5, 10, -7], [0, 0, 6, 0, -1, 0, 0, 0, 0, -4, 0, 0, 0, -1, -5, -6], [-9, 0, 0, -6, 0, 9, -19, 13, 1, 0, 1, 0, 0, 0, 0, 0], [-2, 0, 0, -32, 0, 34, 2, -10, 2, 0, -2, 0, -2, 0, 0, 0], [0, -6, 5, 0, 10, 0, 0, 0, 0, 3, 0, -5, 0, -10, -3, 5], [0, -4, -4, 0, -4, 0, 0, 0, 0, 0, 0, -13, 0, 0, -13, 0], [0, 13, 15, 0, 4, 0, 0, 0, 0, -11, 0, 11, 0, 0, -9, 4], [0, 0, 6, 0, 19, 0, 0, 0, 0, 15, 0, -15, 0, 6, 0, 0], [0, 6, 11, 0, -11, 0, 0, 0, 0, -6, 0, -1, 0, -9, 7, 9], [-6, 0, 0, -30, 0, 1, 0, 23, 3, 0, -3, 0, 1, 0, 0, 0], [7, 0, 0, 15, 0, 6, -5, 7, -7, 0, -5, 0, -7, 0, 0, 0], [0, 0, -8, 0, 5, 0, 0, 0, 0, 18, 0, 0, 0, 5, 1, 8], [-29, 0, 0, -7, 0, -4, -4, 23, -4, 0, -7, 0, 7, 0, 0, 0], [-11, 0, 0, -11, 0, 11, -5, -6, -14, 0, -14, 0, 0, 0, 0, 0], [0, -2, 0, 0, -5, 0, 0, 0, 0, -14, 0, -18, 0, -11, -2, 5], [5, 0, 0, 10, 0, -10, 5, -7, -3, 0, 0, 0, -5, 0, 0, 0], [0, -30, 3, 0, 0, 0, 0, 0, 0, 18, 0, -12, 0, -3, -24, -21], [27, 0, 0, -27, 0, 6, -9, -6, 0, 0, -9, 0, -9, 0, 0, 0], [-11, 0, 0, 28, 0, 0, -6, -11, 6, 0, -2, 0, -2, 0, 0, 0], [0, -10, -14, 0, -14, 0, 0, 0, 0, 0, 0, 4, 0, -6, 4, -6], [0, 2, -15, 0, -10, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, -10], [-25, 0, 0, 15, 0, 16, 17, -16, 0, 0, 7, 0, 17, 0, 0, 0], [18, 0, 0, 8, 0, 21, 0, -11, 5, 0, -5, 0, -7, 0, 0, 0], [12, 0, 0, -24, 0, -12, -6, -3, 3, 0, -6, 0, 3, 0, 0, 0], [0, -27, -5, 0, 0, 0, 0, 0, 0, 6, 0, -6, 0, 0, -15, 0], [17, 0, 0, -29, 0, -17, 7, -36, 0, 0, 0, 0, 0, 0, 0, 0], [4, 0, 0, -20, 0, 0, 7, 4, -7, 0, 2, 0, 2, 0, 0, 0], [0, -24, -9, 0, 0, 0, 0, 0, 0, 26, 0, 2, 0, 9, 4, -5], [-1, 0, 0, -1, 0, 20, -7, -20, 0, 0, -9, 0, -7, 0, 0, 0], [15, 0, 0, -16, 0, -15, -13, -3, -7, 0, -7, 0, 0, 0, 0, 0], [0, 9, 0, 0, 7, 0, 0, 0, 0, 5, 0, 23, 0, 15, 9, -7], [21, 0, 0, 3, 0, 0, -6, 21, 6, 0, -3, 0, -3, 0, 0, 0], [0, 32, 2, 0, 2, 0, 0, 0, 0, 0, 0, 20, 0, 2, 20, 2], [9, 0, 0, 9, 0, -18, -9, -9, 9, 0, 9, 0, -9, 0, 0, 0], [0, -36, -13, 0, 0, 0, 0, 0, 0, 18, 0, -23, 0, 0, -18, -13], [0, 29, -5, 0, 0, 0, 0, 0, 0, -24, 0, 5, 0, 5, 10, 5], [-8, 0, 0, -37, 0, 15, 0, 14, 6, 0, -6, 0, 11, 0, 0, 0], [11, 0, 0, 1, 0, 28, 8, -25, 8, 0, 1, 0, -1, 0, 0, 0], [0, -17, 0, 0, -3, 0, 0, 0, 0, 17, 0, -17, 0, 7, -17, 3], [0, -20, -8, 0, -8, 0, 0, 0, 0, 0, 0, -13, 0, 0, -13, 0], [0, 0, 2, 0, 3, 0, 0, 0, 0, 20, 0, 0, 0, 3, 5, -2], [0, -32, 1, 0, 6, 0, 0, 0, 0, 16, 0, 3, 0, -6, -16, 1], [8, 0, 0, -7, 0, 6, -12, -6, 0, 0, -11, 0, -12, 0, 0, 0], [0, 10, -2, 0, 11, 0, 0, 0, 0, 12, 0, -2, 0, -2, 0, 0], [0, 16, -7, 0, 7, 0, 0, 0, 0, -16, 0, 4, 0, -19, 12, 19], [10, 0, 0, -16, 0, 6, -10, 13, -11, 0, 10, 0, 11, 0, 0, 0], [26, 0, 0, 24, 0, 8, -2, 13, -5, 0, 0, 0, 2, 0, 0, 0], [0, -44, 7, 0, -4, 0, 0, 0, 0, 22, 0, -27, 0, 4, -22, 7], [6, 0, 0, -6, 0, -6, -12, 6, -6, 0, -12, 0, -6, 0, 0, 0], [-3, 0, 0, 23, 0, 3, 5, 18, -2, 0, -2, 0, 0, 0, 0, 0], [0, -1, 0, 0, 3, 0, 0, 0, 0, -8, 0, -10, 0, 11, -1, -3], [-8, 0, 0, 22, 0, 0, 7, -8, -7, 0, -4, 0, -4, 0, 0, 0], [0, -7, -4, 0, -11, 0, 0, 0, 0, 10, 0, -17, 0, -4, 0, 0], [7, 0, 0, 9, 0, -7, -13, 22, 0, 0, 0, 0, 0, 0, 0, 0], [-26, 0, 0, 27, 0, -33, 0, -20, 0, 0, 0, 0, -7, 0, 0, 0], [0, 6, 6, 0, 6, 0, 0, 0, 0, 0, 0, -1, 0, -6, -1, -6], [5, 0, 0, 10, 0, 24, 5, 29, -5, 0, 0, 0, -5, 0, 0, 0], [0, 3, 3, 0, 0, 0, 0, 0, 0, 6, 0, 9, 0, -3, 18, 9], [0, 25, 11, 0, -4, 0, 0, 0, 0, -9, 0, 9, 0, 0, 7, -4], [-8, 0, 0, 13, 0, -14, 9, 16, 9, 0, 13, 0, -13, 0, 0, 0], [-3, 0, 0, -22, 0, 3, 5, -27, 7, 0, 7, 0, 0, 0, 0, 0], [0, -16, -11, 0, 11, 0, 0, 0, 0, 16, 0, -12, 0, 13, -4, -13]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1024_g_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1024_2_g_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_1024_2_g_e(:prec:=16) chi := MakeCharacter_1024_g(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); 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_1024_2_g_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_1024_2_g_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1024_g(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![38416, 0, -34496, 0, 15488, 0, 8288, 0, 2744, 0, -272, 0, 32, 0, 8, 0, 1]>,<5,R![324, 864, 432, -408, 224, -72, 16, -4, 1]>],Snew); return Vf; end function;