// Make newform 900.3.f.i in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_900_f();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_900_f_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_900_3_f_i();". // 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_900_3_f_i();". // 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 := [32761, 0, -53588, 0, 39458, 0, -17800, 0, 5291, 0, -1000, 0, 138, 0, -12, 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], [-92455767, 0, 87520785, 0, -45019766, 0, 12106617, 0, -2294806, 0, 272717, 0, -26035, 0, 1564, 0], [7138059, 0, -11683666, 0, 6222189, 0, -3409951, 0, 769817, 0, -145387, 0, 11182, 0, -1517, 0], [-337443471, 0, 344417086, 0, -173399285, 0, 56040923, 0, -10860033, 0, 1585239, 0, -134882, 0, 12301, 0], [-2555965, 0, 2826986, 0, -1661727, 0, 614137, 0, -128595, 0, 19133, 0, -1702, 0, 159, 0], [4194529, 0, -4798574, 0, 2797615, 0, -1032517, 0, 213667, 0, -32761, 0, 2818, 0, -279, 0], [90006842, 0, -97604685, 0, 49688181, 0, -17610972, 0, 3521601, 0, -538412, 0, 45515, 0, -4499, 0], [10110515, 0, -13936662, 0, 8031169, 0, -2934199, 0, 594605, 0, -90611, 0, 7754, 0, -753, 0], [0, 4520663, 0, -4496945, 0, 2408754, 0, -702033, 0, 125594, 0, -15013, 0, 1455, 0, -76], [0, -1684841, 0, 34028543, 0, -8403134, 0, 9649707, 0, -2166718, 0, 492575, 0, -34121, 0, 5664], [0, -22596309219, 0, 33883360909, 0, -20447352810, 0, 7874858657, 0, -1641213002, 0, 256907341, 0, -21899443, 0, 2233184], [0, 58773711, 0, -66225366, 0, 34130105, 0, -11922963, 0, 2364333, 0, -360879, 0, 30342, 0, -2981], [0, -195043225, 0, 1201016993, 0, -796307516, 0, 319614791, 0, -65146100, 0, 10192219, 0, -867511, 0, 85762], [0, 74953045257, 0, -85332242031, 0, 48923742998, 0, -18269059059, 0, 3781602326, 0, -580754295, 0, 50039697, 0, -4949768], [0, 54030795461, 0, -59181450721, 0, 33485052920, 0, -12429430523, 0, 2544388208, 0, -394763119, 0, 33490087, 0, -3371926], [0, 226052383, 0, -242923901, 0, 130623386, 0, -42962735, 0, 8527534, 0, -1220071, 0, 107187, 0, -9430]]; Rf_basisdens := [1, 6203680, 1772480, 12407360, 129920, 129920, 886240, 129920, 839840, 11757760, 4491464320, 5878880, 154878080, 4491464320, 2245732160, 2939440]; 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_900_f();" function MakeCharacter_900_f() N := 900; order := 2; char_gens := [451, 101, 577]; v := [1, 2, 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_900_f_Hecke();" function MakeCharacter_900_f_Hecke(Kf) N := 900; order := 2; char_gens := [451, 101, 577]; char_values := [[-1, 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], [-1, 0, 0, 0, 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 := 3; raw_aps := [[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, 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, -1, 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, 0, -1, 2, 0, -1, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, -2, 3, 0], [-1, 0, 0, 0, -2, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, -1, -1, 0], [0, 1, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 4, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 2, 0, 17, 0, 0, 0, -1], [0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, -4, 0, -2, 0, 0, 0, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, -8, 1, -3, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, -7, -4, 0], [0, -6, -6, 18, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 6, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -6, -8, 0, -4, 0, 0, 0, -4], [0, -2, -2, 6, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, -4, 0, 15, 0, 0, 0, 2], [-3, 0, 0, 0, 4, 19, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -40, 0, 4, -8, -6, 0], [0, -26, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -6, 12, 0, 15, 0, 0, 0, -6], [0, -11, -16, -16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -9, 4, 0, 2, 0, 0, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 20, 2, 12, 0], [94, 0, 0, 0, 0, 12, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22, 0, 0, 8, -15, 0], [0, 0, 0, 0, 0, 0, 0, 0, 17, -12, 0, -6, 0, 0, 0, -6], [0, -10, -10, 30, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 28, 0, 0, -30, 1, 0], [1, 0, 0, 0, 6, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 11, -30, -30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 18, 23, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 6, 0, 9, 0, 0, 0, -3], [0, 0, 0, 0, 0, 0, 0, 0, -8, 4, 0, 2, 0, 0, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -15, 0, 0, -3, -3, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 73, 0, 0, -61, -6, 0], [0, -14, -14, 42, 0, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-6, 0, 0, 0, 0, -12, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, -8, -8, 24, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, -8, 0, -55, 0, 0, 0, 4], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -47, 0, 0, 19, 14, 0], [5, 0, 0, 0, 2, -23, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0], [7, 0, 0, 0, -10, -45, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 27, -20, 0, -10, 0, 0, 0, -10], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, 0, -4, 0], [-44, 0, 0, 0, 0, -6, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -38, 0, 0, 32, 3, 0], [0, -14, -14, 42, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [178, 0, 0, 0, 0, 36, 0, -12, 0, 0, 0, 0, 0, 0, 0, 0], [0, 8, 8, -24, 0, 0, -19, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 42, 0, 0, -8, -17, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -55, 0, 32, -11, 5, 0], [0, 25, 42, 42, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 0, 0, 0, 18, 43, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 21, -42, 0, 67, 0, 0, 0, 21], [0, 80, -14, -14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 30, 24, 0, 12, 0, 0, 0, 12], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 67, 0, 0, 1, -34, 0], [0, 0, 0, 0, 0, 0, 0, 0, 14, 12, 0, 6, 0, 0, 0, 6], [0, -6, -6, 18, 0, 0, -26, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -16, 32, 0, -63, 0, 0, 0, -16], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -103, 0, 0, 67, 18, 0], [-1, 0, 0, 0, 14, 19, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, -4, 0, -219, 0, 0, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 0, -40, -2, -22, 0], [68, 0, 0, 0, 0, -18, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 24, -13, 0], [0, 26, 26, -78, 0, 0, -28, 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, 23, -46, 0, 7, 0, 0, 0, 23], [19, 0, 0, 0, 66, -29, 0, -19, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 115, 0, -16, 23, 15, 0], [0, 41, -24, -24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -23, 46, 0, 71, 0, 0, 0, -23], [0, -12, -6, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-70, 0, 0, 0, 0, 36, 0, -12, 0, 0, 0, 0, 0, 0, 0, 0], [0, 20, 20, -60, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0], [176, 0, 0, 0, 0, -126, 0, 42, 0, 0, 0, 0, 0, 0, 0, 0], [0, 38, 38, -114, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -6, 12, 0, 51, 0, 0, 0, -6], [5, 0, 0, 0, -26, -51, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 60, 0, 80, 12, 52, 0], [0, 138, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -6, 12, 0, -113, 0, 0, 0, -6], [0, -59, -34, -34, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 11, 48, 0, 24, 0, 0, 0, 24], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, -44, 2, -20, 0], [0, 30, 30, -90, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -17, -36, 0, -18, 0, 0, 0, -18], [0, -18, -18, 54, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-15, 0, 0, 0, -6, 69, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 195, 0, 40, 39, 59, 0], [0, 95, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 30, 68, 68, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 22, -24, 0, -12, 0, 0, 0, -12], [-98, 0, 0, 0, 0, 96, 0, -32, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 6, -44, 0, -22, 0, 0, 0, -22], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 51, 0, 0, 65, -58, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 170, 0, -112, 34, -22, 0], [0, -72, 62, 62, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [11, 0, 0, 0, 86, 31, 0, -11, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -22, 44, 0, -69, 0, 0, 0, -22], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 36, 2, 20, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 106, 0, 0, -60, -23, 0], [0, 30, 30, -90, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0], [362, 0, 0, 0, 0, -12, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -83, 8, 0, 4, 0, 0, 0, 4], [0, 0, 0, 0, 0, 0, 0, 0, -29, 58, 0, 209, 0, 0, 0, -29], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 106, 0, 0, -52, -27, 0], [-27, 0, 0, 0, 22, 157, 0, 27, 0, 0, 0, 0, 0, 0, 0, 0], [19, 0, 0, 0, -40, -135, 0, -19, 0, 0, 0, 0, 0, 0, 0, 0], [0, -150, 58, 58, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 24, -4, 0, -2, 0, 0, 0, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 295, 0, -24, 59, 47, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -91, 0, 0, 27, 32, 0], [0, 30, 30, -90, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-348, 0, 0, 0, 0, 90, 0, -30, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -14, 28, 0, 111, 0, 0, 0, -14], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 251, 0, 0, -75, -88, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 190, 0, 52, 38, 64, 0], [-9, 0, 0, 0, 58, 103, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0], [0, 73, 88, 88, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [852, 0, 0, 0, 0, -6, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 36, 36, -108, 0, 0, -20, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -65, -36, 0, -18, 0, 0, 0, -18], [0, 0, 0, 0, 0, 0, 0, 0, -5, 10, 0, -175, 0, 0, 0, -5], [13, 0, 0, 0, -58, -123, 0, -13, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -95, 0, 144, -19, 53, 0], [17, 0, 0, 0, -32, -117, 0, -17, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -23, 46, 0, 399, 0, 0, 0, -23], [0, -2, -32, -32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-458, 0, 0, 0, 0, -84, 0, 28, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -357, 0, 0, 209, 74, 0], [0, 0, 0, 0, 0, 0, 0, 0, 90, 16, 0, 8, 0, 0, 0, 8], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -129, 0, 0, -75, 102, 0], [0, 170, -70, -70, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [15, 0, 0, 0, 98, 23, 0, -15, 0, 0, 0, 0, 0, 0, 0, 0], [0, 39, 146, 146, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -75, 68, 0, 34, 0, 0, 0, 34], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 140, 0, 72, 28, 64, 0], [438, 0, 0, 0, 0, 156, 0, -52, 0, 0, 0, 0, 0, 0, 0, 0], [0, -6, -6, 18, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -25, 50, 0, 573, 0, 0, 0, -25], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 160, 0, 0, -118, -21, 0], [-17, 0, 0, 0, 114, 199, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -185, 0, -96, -37, -85, 0], [0, 0, 0, 0, 0, 0, 0, 0, 37, -74, 0, 157, 0, 0, 0, 37], [0, -110, 12, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -108, 68, 0, 34, 0, 0, 0, 34], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -155, 0, -88, -31, -75, 0], [0, 0, 0, 0, 0, 0, 0, 0, 128, -40, 0, -20, 0, 0, 0, -20], [0, -38, -38, 114, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 0, 0, 0, -76, -51, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 134, 20, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 28, -56, 0, -627, 0, 0, 0, 28], [0, -73, -56, -56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 210, 0, 56, 42, 70, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 292, 0, 0, 6, -149, 0], [0, 0, 0, 0, 0, 0, 0, 0, 117, 20, 0, 10, 0, 0, 0, 10], [0, 24, 24, -72, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 140, 0, 0, 158, -149, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 45, 0, -208, 9, -95, 0], [-23, 0, 0, 0, 10, 125, 0, 23, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 11, -22, 0, 295, 0, 0, 0, 11]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_900_f_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_900_3_f_i();". // 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_900_3_f_i(:prec:=16) chi := MakeCharacter_900_f(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 3)); 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_900_3_f_i();". // 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_900_3_f_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_900_f(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign))); Vf := Kernel([<7,R![464, 0, -136, 0, 1]>,<29,R![35344, 0, -456, 0, 1]>],Snew); return Vf; end function;