// Make newform 6240.2.a.bq in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6240_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_6240_2_a_bq();". // 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_6240_2_a_bq();". // 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, -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_6240_a();" function MakeCharacter_6240_a() N := 6240; order := 1; char_gens := [1951, 2341, 2081, 2497, 5761]; v := [1, 1, 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_6240_a_Hecke(Kf) return MakeCharacter_6240_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 := [[0, 0], [1, 0], [-1, 0], [0, -1], [0, -1], [1, 0], [-2, 1], [-4, 2], [4, 1], [-6, 2], [-4, 4], [2, -3], [-2, -3], [-4, -2], [8, 0], [-2, 1], [-4, 0], [2, 3], [-4, 4], [-4, 3], [-6, -4], [-8, 1], [-4, 0], [-2, -1], [-2, 3], [2, -2], [4, -8], [0, 1], [-10, 2], [2, 0], [12, 0], [4, -8], [-6, 0], [-8, 3], [6, -9], [4, -8], [-10, -4], [-16, -1], [8, 0], [-6, 4], [-4, 2], [2, -1], [8, 6], [-10, -3], [2, -8], [4, -4], [4, 4], [4, -4], [4, -2], [-2, 10], [-10, 3], [12, -5], [-6, 6], [4, 0], [-14, 4], [8, 0], [10, -6], [-4, -6], [-18, 2], [2, -4], [-4, 4], [-22, 2], [-8, -1], [0, 2], [-6, 4], [-22, 0], [-20, 2], [18, -4], [-8, -3], [-26, 4], [2, 2], [16, 0], [12, 6], [-2, -12], [-4, 0], [-8, -2], [-6, 14], [-6, -3], [2, 12], [-14, 14], [20, 2], [-18, 2], [-8, 16], [18, 0], [-16, -1], [8, 5], [-26, 1], [-10, 15], [6, 1], [-16, 7], [0, -3], [12, -15], [24, -3], [4, 14], [20, -10], [0, 0], [22, -9], [18, -8], [-4, -10], [14, 4], [28, 0], [-22, 6], [16, -7], [-14, 4], [-32, -3], [14, -13], [4, 14], [2, -14], [16, 8], [30, -7], [4, -18], [-22, 3], [-30, -4], [4, -16], [20, 4], [42, 0], [-16, 3], [28, -1], [26, -8], [-44, 0], [-2, -16], [10, 2], [22, -11], [12, -10], [12, 6], [10, 14], [-18, -2], [-8, 0], [36, -2], [-30, -3], [-28, 6], [32, -10], [16, -1], [-34, 2], [10, 12], [18, -16], [-6, 2], [-12, 4], [-26, -1], [-30, 8], [4, -14], [38, -3], [12, -16], [-4, -12], [-18, 20], [-4, 7], [-6, 15], [30, -1], [-16, 15], [8, 10], [-26, 20], [26, -2], [-36, 0], [-28, 11], [-4, -10], [-8, -6], [0, -11], [14, -13], [-14, -6], [-2, -5], [-12, 12], [22, -3], [28, -4], [12, -22], [18, 4], [40, -8], [0, -13], [-26, 10], [10, -2], [38, -7], [-4, 16], [22, -12], [48, 2], [6, -25], [-16, 11], [10, -12], [-28, 4], [38, 3], [-12, 28], [38, -8], [-8, 15], [-12, 6], [-18, -14], [-54, 2], [-8, 18], [10, 10], [-34, -8], [16, -7], [26, 0], [-12, 1], [50, 4], [-28, 10], [24, -15], [22, -23], [-12, -8], [-34, -7], [-30, 24], [38, 10], [-14, -18], [12, -19], [-14, 0], [16, -7], [18, 7], [-2, 15], [8, 1], [30, -9], [-4, -6], [-20, 16], [-18, 23], [0, 3], [-6, 14], [-6, 6], [-44, 0], [-20, -12], [-32, 8], [10, 24], [12, -18], [2, 2], [16, 10], [-46, -8], [10, 21], [4, -2], [-30, 8], [-28, 8], [16, -27], [-22, 11], [6, -17], [8, 0], [12, 2], [-8, -5], [-18, -12], [24, -1], [-36, -6], [50, 10], [28, -2], [24, 4], [18, 0], [58, 0], [12, 2], [40, -6], [44, 4], [40, -1], [-20, 16], [38, -20], [26, -4], [-24, 2], [-20, 20], [16, 3], [-12, -24], [-48, -8], [-18, 16], [42, 8], [48, -4], [-42, 7], [-34, 19], [40, 9], [6, -12], [-16, 7], [10, 0], [-10, -5], [4, -10], [48, -9], [-18, -6], [6, 8], [-38, -6], [40, 9], [14, -1], [-18, 15], [-16, -11], [2, -28], [6, -4], [32, 9], [38, 3], [-12, 16], [10, -18], [-48, 5], [-76, 4], [-2, -22], [26, -24], [-4, 2], [12, -7], [-52, 20], [-32, 24], [54, -2], [-36, 16], [16, -20], [18, -8], [-30, 30], [-28, 0], [-6, 22], [26, -28], [-12, 0], [-6, -10], [-52, 8], [-10, 32], [26, 20], [-4, 8], [-10, -9], [44, -10], [-16, -5], [18, -10], [2, 24], [-8, -11], [16, -29], [4, -8], [70, 1], [0, 7], [-14, -7], [32, -8], [22, -2], [-44, -3], [50, 12], [-14, 14], [20, -6], [48, -2], [6, 3], [-8, 13], [-28, -1], [-46, 20], [-6, -18], [36, -20], [-54, 0], [-46, 20], [-40, -9], [-46, -2], [-10, 17], [-12, 8], [-8, -13], [36, -27], [18, 8], [46, -8], [-58, -9], [-20, 4], [12, -26], [52, 4], [-76, 0], [-50, 10], [2, -6], [2, 6], [-28, 6], [-22, 3], [-34, -9], [46, 7], [52, -4], [-6, 20], [20, -2], [-30, 7], [64, -13], [8, 16], [38, 7], [20, 20], [-50, 13], [-18, 21], [-44, 22], [2, -7], [-70, -8], [-36, 39], [-20, 8], [-22, 36], [0, -10], [-34, -8], [42, -4], [-16, -24], [52, 0], [28, 10], [18, -10], [34, -2], [64, -17], [-26, 7], [28, -28], [20, 8], [-24, -2], [-70, 12], [-24, -13], [-18, -28], [16, -11], [-16, 28], [-34, -5], [-30, -2], [26, 0], [2, 20], [-70, 0], [20, 12], [-22, -4], [68, -4], [40, 4], [84, 6], [-26, -16], [20, 8], [-68, 17], [50, -32], [34, 10], [24, 3], [36, -22], [-60, 3], [-22, 32], [-36, -10], [58, -18], [16, 19], [-22, 32], [-42, 16], [-26, 27], [24, 13], [-6, 28], [-10, 21], [8, 11], [14, -20], [-30, 12], [-56, -7], [0, -15], [50, 4], [14, 23], [24, 21], [8, 31], [18, -34], [-54, -2], [-28, -15], [-20, -20], [-54, -6], [12, 15], [-62, 2], [2, -27], [-24, -26], [52, -28], [26, 24], [-14, -18], [-16, -15], [-26, 33], [-36, 4], [-40, -14]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6240_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6240_2_a_bq();". // 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_6240_2_a_bq(:prec:=2) chi := MakeCharacter_6240_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(2999) - 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_6240_2_a_bq();". // 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_6240_2_a_bq( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6240_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![-4, 1, 1]>,<11,R![-4, 1, 1]>,<17,R![-2, 3, 1]>,<19,R![-8, 6, 1]>],Snew); return Vf; end function;