// Make newform 450.4.c.e in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_450_c();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_450_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_450_4_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_450_4_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 := [1, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0], [0, 2]]; Rf_basisdens := [1, 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_450_c();" function MakeCharacter_450_c() N := 450; order := 2; char_gens := [101, 127]; v := [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_450_c_Hecke();" function MakeCharacter_450_c_Hecke(Kf) N := 450; order := 2; char_gens := [101, 127]; char_values := [[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 := 4; raw_aps := [[0, -1], [0, 0], [0, 0], [0, 8], [-12, 0], [0, 19], [0, -63], [-20, 0], [0, -84], [30, 0], [-88, 0], [0, -127], [-42, 0], [0, -26], [0, -48], [0, -99], [-660, 0], [-538, 0], [0, -442], [-792, 0], [0, 109], [520, 0], [0, 246], [810, 0], [0, -577], [618, 0], [0, 64], [0, -738], [-1190, 0], [0, 231], [0, 1268], [-2292, 0], [0, -363], [-380, 0], [1590, 0], [2432, 0], [0, -307], [0, -926], [0, -1068], [0, -879], [-540, 0], [1982, 0], [2688, 0], [0, -1151], [0, 2187], [1600, 0], [3332, 0], [0, 1324], [0, 1122], [5650, 0], [0, -2349], [-1200, 0], [-718, 0], [-6012, 0], [0, -1023], [0, 3036], [-6930, 0], [1352, 0], [0, 593], [-2442, 0], [0, 1414], [0, -2379], [0, 4238], [-4632, 0], [0, -2411], [0, -1713], [-2788, 0], [0, -217], [0, 3342], [-2630, 0], [0, 3711], [-10440, 0], [0, -5212], [0, 1639], [-6140, 0], [0, 1536], [6150, 0], [0, 53], [1758, 0], [3670, 0], [-9660, 0], [8462, 0], [-9792, 0], [0, -3671], [-10640, 0], [0, 8706], [-1710, 0], [0, 323], [6018, 0], [0, -3356], [0, 2682], [9840, 0], [0, -712], [4548, 0], [-6500, 0], [0, -6084], [-21090, 0], [5238, 0], [0, 4294], [3062, 0], [0, 4238], [0, -6273], [0, 6], [19290, 0], [-12148, 0], [0, 5183], [0, 3822], [0, -4329], [25800, 0], [16202, 0], [0, 12068], [0, -2321], [0, -3363], [21220, 0], [29792, 0], [10158, 0], [0, 14914], [0, 972], [0, -13359], [4260, 0], [22862, 0], [0, -16271], [0, 7107], [0, 3546], [-13228, 0], [-28062, 0], [27250, 0], [-14400, 0], [0, -8992], [0, 8299], [-1460, 0], [0, 15036], [-18088, 0], [0, -12367], [22278, 0], [-16130, 0], [0, -14859], [0, -4762], [0, -16953], [-630, 0], [-20788, 0], [43098, 0], [0, -7136], [0, 6822], [2410, 0], [23160, 0], [0, 16039], [0, -7203], [-30620, 0], [0, -8784], [0, 10853], [14958, 0], [0, -16406], [0, -19428], [0, 14138], [-8112, 0], [26080, 0], [49170, 0], [0, -24157], [-34782, 0], [0, -12558], [0, 7731], [0, 368], [29268, 0], [0, 8337], [0, 15636], [-15928, 0], [0, -21007]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_450_c_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_450_4_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_450_4_c_e(:prec:=2) chi := MakeCharacter_450_c(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 4)); 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_450_4_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_450_4_c_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_450_c(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<7,R![256, 0, 1]>,<11,R![12, 1]>],Snew); return Vf; end function;