// Make newform 3920.2.a.bg in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3920_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_3920_2_a_bg();". // 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_3920_2_a_bg();". // 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_3920_a();" function MakeCharacter_3920_a() N := 3920; order := 1; char_gens := [1471, 981, 3137, 3041]; 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_3920_a_Hecke(Kf) return MakeCharacter_3920_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], [2], [-1], [0], [4], [-2], [-8], [-6], [4], [-6], [4], [-10], [-4], [-4], [4], [10], [-14], [10], [4], [-12], [-4], [-4], [-2], [-8], [0], [2], [4], [12], [10], [-2], [12], [18], [-2], [-10], [-10], [0], [-2], [4], [12], [-18], [-4], [-26], [-12], [-18], [18], [4], [-20], [-8], [-18], [14], [22], [8], [-4], [22], [-12], [0], [-26], [16], [2], [-10], [-26], [-6], [-2], [16], [8], [18], [28], [-14], [4], [-10], [12], [8], [8], [-6], [4], [20], [18], [-26], [-14], [20], [6], [-34], [0], [40], [-32], [-4], [18], [6], [6], [-16], [-10], [4], [44], [12], [-28], [-24], [6], [12], [-34], [-34], [-28], [-6], [18], [-38], [36], [20], [2], [20], [28], [-40], [-16], [-2], [-6], [-6], [-4], [18], [18], [28], [14], [4], [2], [22], [46], [12], [-46], [-38], [42], [36], [20], [-30], [-12], [12], [-40], [-2], [12], [-28], [18], [6], [-2], [-10], [2], [-18], [40], [-44], [-14], [-36], [-50], [24], [38], [24], [-42], [40], [20], [4], [-4], [-40], [4], [12], [-12], [10], [12], [2], [52], [-62], [54], [20], [36], [42], [-2], [14], [36], [-26], [36], [-42], [4], [-48], [-20], [26], [8], [50], [-28], [-18], [22], [-12], [-44], [30], [26], [-26], [6], [-36], [-36], [36], [-36], [-42], [-52], [-12], [10], [46], [68], [-48], [-66], [44], [-14], [8], [42], [-6], [-20], [-60], [-2], [-2], [-14], [18], [-4], [38], [32], [-28], [-32], [-20], [48], [-34], [18], [-16], [-30], [20], [22], [-38], [36], [-48], [8], [-36], [30], [38], [-20], [26], [-34], [-32], [-4], [-30], [-36], [-40], [36], [-14], [72], [14], [44], [20], [-48], [-46], [20], [48], [-6], [-24], [56], [-64], [26], [-76], [-54], [-78], [-46], [-24], [-60], [-28], [-62], [-54], [8], [30], [70], [0], [-36], [-66], [-50], [-52], [28], [-56], [-28], [72], [4], [46], [-2], [34], [-24], [-12], [52], [58], [38], [-16], [-38], [-14], [-68], [-8], [46], [30], [-38], [34], [-38], [6], [-48], [6], [22], [-70], [32], [-58], [52], [-36], [4], [-10], [-60], [-54], [-48], [-22], [40], [10], [-82], [60], [24], [-8], [-18], [-68], [4], [46], [-50], [62], [-78], [-56], [-26], [-4], [4], [-74], [44], [-2], [-82], [-34], [-4], [-30], [-20], [54], [-6], [84], [-16], [52], [82], [42], [-58], [36], [38], [12], [-26], [28], [-24], [-70], [30], [-62], [30], [52], [-26], [-36], [20], [66], [42], [68], [42], [48], [12], [-44], [38], [26], [-42], [-52], [22], [-84], [54], [32], [6], [64], [-54], [34], [40], [-4], [68], [-8], [22], [-74], [16], [2], [-82], [-48], [-20], [-70], [-92], [-12], [10], [26], [4], [-62], [-48], [-18], [-44], [-28], [100], [50], [-54], [42], [-80], [-32], [-10], [-36], [-54], [62], [50], [54], [4], [62], [-36], [-28], [-58], [-58], [-20], [-60], [88], [-40], [-6], [98], [16], [-42], [24], [10], [28], [38], [70], [36]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3920_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3920_2_a_bg();". // 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_3920_2_a_bg(:prec:=1) chi := MakeCharacter_3920_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_3920_2_a_bg();". // 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_3920_2_a_bg( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3920_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-2, 1]>,<11,R![-4, 1]>,<13,R![2, 1]>,<17,R![8, 1]>],Snew); return Vf; end function;