// Make newform 2880.2.a.bh in Magma, downloaded from the LMFDB on 30 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2880_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_2880_2_a_bh();". // 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_2880_2_a_bh();". // 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_2880_a();" function MakeCharacter_2880_a() N := 2880; order := 1; char_gens := [2431, 901, 641, 577]; 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_2880_a_Hecke(Kf) return MakeCharacter_2880_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], [4], [0], [2], [6], [0], [-4], [-2], [8], [-6], [6], [12], [-12], [-10], [-8], [10], [-12], [8], [10], [-16], [-12], [6], [18], [6], [-4], [-4], [-6], [6], [4], [0], [14], [0], [-10], [-8], [18], [-4], [-12], [14], [24], [2], [-8], [2], [6], [-16], [8], [28], [28], [2], [-18], [0], [-14], [16], [22], [-4], [-2], [8], [-22], [-10], [-4], [6], [4], [-8], [-22], [30], [-8], [2], [-20], [-6], [6], [0], [-12], [-6], [-16], [-4], [-26], [-14], [-18], [-22], [40], [18], [-24], [18], [16], [4], [-18], [-22], [30], [-20], [-36], [0], [4], [-16], [32], [-36], [-2], [-26], [28], [-38], [4], [-18], [4], [38], [-40], [34], [12], [-26], [-16], [-6], [-12], [-38], [14], [-16], [8], [30], [-36], [4], [-50], [-24], [-14], [34], [6], [36], [40], [-18], [-30], [-48], [20], [2], [-16], [-4], [-24], [-22], [22], [-46], [6], [-28], [30], [-10], [-40], [6], [-4], [-20], [10], [-16], [10], [46], [32], [28], [2], [-18], [-20], [-28], [4], [24], [-16], [14], [-38], [30], [-4], [-34], [4], [0], [6], [44], [-40], [42], [2], [-42], [-24], [10], [40], [-22], [32], [-58], [40], [-42], [-4], [10], [-28], [0], [26], [30], [44], [6], [18], [28], [-6], [-24], [18], [4], [56], [30], [60], [-34], [34], [-14], [-42], [28], [62], [56], [26], [2], [-40], [30], [0], [52], [22], [-8], [-14], [-42], [12], [-36], [-64], [26], [36], [-18], [-60], [-34], [18], [-32], [-18], [-4], [28], [34], [-18], [64], [52], [-48], [-62], [-16], [40], [38], [12], [36], [-14], [54], [-24], [-8], [36], [-56], [-52], [42], [54], [-16], [36], [-64], [16], [12], [66], [-66], [20], [-54], [46], [-8], [-62], [-28], [6], [-22], [12], [-68], [34], [34], [38], [32], [-34], [22], [44], [-58], [-38], [-12], [58], [-32], [2], [-4], [-36], [26], [-22], [32], [-36], [8], [84], [-30], [68], [72], [34], [-10], [-32], [-18], [-50], [60], [-2], [-64], [50], [30], [8], [54], [40], [20], [-38], [-50], [0], [4], [40], [-78], [12], [-38], [48], [74], [44], [-10], [30], [-36], [68], [-70], [40], [-56], [50], [-34], [-88], [-22], [14], [76], [-34], [-78], [-80], [-4], [84], [38], [-54], [-2], [-16], [-44], [-40], [-36], [-70], [70], [42], [20], [26], [46], [54], [-8], [-66], [8], [50], [-12], [-56], [-42], [8], [-22], [62], [-52], [-30], [30], [48], [-16], [-10], [-84], [10], [-26], [-60], [24], [-28], [-70], [-82], [-52], [10], [48], [16], [-100], [6], [88], [2], [40], [24], [2], [-2], [26], [78], [14], [-60], [-26], [-16], [60], [-40], [-22], [-4], [-28], [-46], [70], [-40], [68], [24], [26], [-32], [54], [-24], [-42], [-70], [-26], [20], [-2], [86], [104], [-78], [-50], [76], [-24], [-14], [-42], [-28], [24], [106], [-18], [0], [-12], [-74], [-52], [-34], [-54], [20], [-56], [-38], [14], [100], [38], [56], [96]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2880_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2880_2_a_bh();". // 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_2880_2_a_bh(:prec:=1) chi := MakeCharacter_2880_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_2880_2_a_bh();". // 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_2880_2_a_bh( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2880_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![-4, 1]>,<11,R![0, 1]>,<13,R![-2, 1]>,<17,R![-6, 1]>,<19,R![0, 1]>],Snew); return Vf; end function;