// Make newform 525.2.a.e in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_525_a();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_525_2_a_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_525_2_a_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, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]]; 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_525_a();" function MakeCharacter_525_a() N := 525; order := 1; char_gens := [176, 127, 451]; v := [1, 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; function MakeCharacter_525_a_Hecke(Kf) return MakeCharacter_525_a(); 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 := [[-1, -1], [-1, 0], [0, 0], [1, 0], [1, -4], [-4, 2], [-2, 6], [0, -2], [-5, 0], [-5, 6], [-2, 4], [1, -4], [-8, 0], [-5, -2], [-4, -2], [6, -4], [-4, 2], [4, -12], [-7, 6], [-7, 4], [2, -2], [5, -2], [-6, -4], [4, -6], [-6, -4], [-14, 6], [0, -6], [-8, 8], [13, -12], [13, 2], [9, 2], [-14, 6], [-6, 4], [-6, 6], [5, 2], [11, -6], [-4, -8], [-12, -4], [16, -6], [20, -10], [-8, 8], [-18, 14], [8, -12], [19, -12], [9, -2], [-16, 0], [0, 8], [-2, 2], [8, -2], [-10, 20], [-17, 6], [4, -20], [16, 2], [6, -2], [2, 4], [-15, 8], [-22, -4], [20, 2], [-10, 16], [-9, 14], [-6, 12], [4, 14], [-2, -14], [10, 2], [-4, 6], [5, -14], [17, -2], [22, 4], [13, 0], [4, -2], [-14, 8], [13, 8], [-4, 8], [15, 0], [-19, 22], [4, -18], [-5, 2], [18, 0], [-17, -6], [8, -10], [22, -8], [19, -12], [8, 16], [8, -28], [-8, -2], [-32, 12], [13, -6], [23, 4], [-12, 14], [4, -20], [2, -16], [-20, -2], [23, -10], [-7, 4], [20, -32], [2, -12], [-8, -14], [14, -32], [-16, 32], [-5, -8], [-7, 26], [-15, 14], [10, -28], [-9, -2], [-31, -6], [-8, -2], [-22, 18], [20, -28], [-33, -8], [4, -20], [-12, 6], [-23, 8], [-1, 14], [-4, 18], [-11, -18], [-25, 6], [-20, 40], [14, -8], [-6, -12], [28, 8], [-8, 10], [6, 0], [2, -28], [-31, 16], [20, -20], [14, 12], [18, 16], [-16, 36], [-30, 30], [-12, 0], [7, -2], [-28, 36], [8, 20], [1, 8], [-36, 28], [16, 12], [22, 8], [-42, 2], [-2, 26], [-23, 10], [-6, 0], [2, -8], [-11, 34], [-19, -8], [30, -2], [-2, 18], [-20, -20], [-20, -12], [6, -4], [-15, 8], [-6, 0], [-6, -20], [-31, -2], [-4, 12], [-16, 20], [27, -12], [-19, 34], [-14, -10], [28, -20], [6, 24], [-20, -8], [13, -18], [28, -16], [-24, 22], [-13, 6], [22, -12], [-27, 22], [-4, -30]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_525_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_525_2_a_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_525_2_a_e(:prec:=2) chi := MakeCharacter_525_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); 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_525_2_a_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_525_2_a_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_525_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![1, 3, 1]>,<11,R![-19, 2, 1]>],Snew); return Vf; end function;