// Make newform 6930.2.a.g in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6930_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_6930_2_a_g();". // 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_6930_2_a_g();". // 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 Kf := Rationals(); end if; return [Kf!elt[1] : elt in input]; end function; // To make the character of type GrpDrchElt, type "MakeCharacter_6930_a();" function MakeCharacter_6930_a() N := 6930; order := 1; char_gens := [1541, 1387, 2971, 2521]; v := [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_6930_a_Hecke(Kf) return MakeCharacter_6930_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], [0], [-1], [1], [1], [4], [4], [8], [-2], [6], [-4], [4], [-2], [4], [4], [-2], [0], [10], [14], [-8], [-10], [4], [4], [-16], [-14], [6], [-16], [-12], [-4], [-4], [0], [12], [-4], [12], [10], [12], [6], [14], [-18], [18], [4], [-8], [-20], [-14], [-10], [12], [6], [-24], [4], [-12], [14], [6], [2], [-24], [14], [20], [14], [16], [10], [4], [-14], [14], [-14], [30], [6], [6], [-20], [-34], [12], [-14], [6], [-14], [-24], [6], [-20], [12], [14], [-30], [6], [14], [28], [-2], [14], [-14], [40], [12], [6], [10], [-14], [16], [-36], [-24], [0], [28], [-8], [30], [30], [32], [-42], [16], [-32], [-10], [-24], [-40], [18], [6], [36], [32], [-12], [14], [32], [-6], [4], [46], [8], [-18], [12], [-36], [10], [-36], [40], [-50], [26], [36], [-26], [-2], [30], [42], [48], [-20], [34], [-16], [32], [-28], [6], [-2], [-26], [-30], [-34], [-48], [56], [6], [4], [-52], [16], [54], [4], [0], [50], [-34], [2], [-40], [-2], [-6], [-38], [16], [-48], [12], [-22], [-10], [-16], [-30], [8], [36], [-32], [12], [8], [-16], [-22], [46], [8], [-24], [-30], [10], [-40], [24], [-30], [54], [28], [-14], [-8], [60], [-52], [36], [46], [-30], [6], [20], [26], [-24], [-2], [12], [32], [-6], [12], [10], [50], [36], [-48], [-14], [50], [-56], [-50], [30], [-40], [-2], [20], [4], [-44], [-62], [50], [2], [-44], [-28], [-20], [30], [32], [-42], [-58], [-62], [-52], [24], [-50], [4], [24], [4], [18], [48], [8], [-12], [-32], [-20], [44], [-60], [64], [14], [-30], [-6], [36], [6], [60], [0], [40], [-74], [-16], [8], [-40], [-68], [38], [-36], [2], [-50], [-62], [-18], [70], [48], [14], [-58], [-58], [-26], [32], [-36], [10], [12], [-38], [26], [-18], [12], [56], [-6], [64], [-58], [-14], [-64], [-50], [-32], [0], [-16], [34], [36], [-26], [-16], [66], [58], [74], [0], [-46], [58], [-80], [-38], [50], [-12], [-26], [-76], [66], [-30], [-8], [50], [-84], [70], [38], [-50], [-52], [68], [24], [-62], [-16], [28], [-16], [-42], [10], [-30], [44], [-30], [-68], [34], [36], [-18], [-86], [12], [16], [22], [42], [-64], [-6], [-14], [-40], [20], [16], [-70], [-48], [18], [-8], [24], [-26], [-84], [-58], [64], [6], [52], [-64], [36], [-26], [-48], [-42], [72], [-44], [74], [36], [6], [48], [50], [-18], [4], [-24], [-56], [26], [84], [-62], [18], [-14], [34], [-22], [72], [20], [42], [-74], [-56], [-42], [44], [-6], [72], [46], [-88], [32], [76], [30], [-38], [-6], [-14], [-70], [-68], [72], [-82], [-20], [76], [96], [-82], [76], [56], [62], [-38], [36], [-32], [16], [58], [24], [40], [-92], [-78], [78], [8], [88], [-42], [18], [32], [-104], [-88], [-4], [8], [74], [-54], [-52], [2], [70], [70], [-42], [-56], [50], [-26], [30], [92], [82], [-36], [-10], [14], [36], [40], [-50], [84], [10], [-44], [-56], [-12], [-2], [-96], [46], [74], [2], [64], [60], [42], [-92], [-14], [-110], [42], [86], [-46], [62], [26], [12], [58], [76], [-86], [38], [10], [28], [-12], [8], [-12], [32], [64], [-72], [70], [38], [46], [-24], [4], [-96], [42], [-32], [64], [-54], [38], [-20], [20], [-94], [-40], [-58], [66], [74], [-70]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6930_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6930_2_a_g();". // 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_6930_2_a_g(:prec:=1) chi := MakeCharacter_6930_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(3449) - 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_6930_2_a_g();". // 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_6930_2_a_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6930_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<13,R![-4, 1]>,<17,R![-4, 1]>,<19,R![-8, 1]>,<23,R![2, 1]>,<29,R![-6, 1]>,<31,R![4, 1]>],Snew); return Vf; end function;