// Make newform 3120.2.a.k in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3120_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_3120_2_a_k();". // 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_3120_2_a_k();". // 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_3120_a();" function MakeCharacter_3120_a() N := 3120; order := 1; char_gens := [1951, 2341, 2081, 2497, 2641]; 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_3120_a_Hecke(Kf) return MakeCharacter_3120_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], [-1], [1], [0], [-4], [1], [2], [4], [-8], [-2], [8], [6], [-6], [4], [8], [6], [12], [-2], [4], [0], [-6], [-16], [4], [10], [18], [6], [8], [-12], [-2], [2], [16], [12], [-6], [20], [-10], [0], [-2], [20], [-16], [-2], [12], [22], [-16], [-14], [6], [8], [-20], [24], [-12], [22], [26], [24], [-14], [4], [18], [24], [14], [8], [-10], [-22], [20], [6], [4], [-24], [26], [30], [-20], [-14], [4], [14], [-14], [-16], [16], [22], [-36], [-24], [6], [14], [-30], [-38], [-20], [-10], [-8], [18], [24], [4], [18], [26], [-18], [-8], [-4], [-8], [-16], [20], [-28], [-24], [30], [-22], [20], [14], [-36], [-18], [-4], [-38], [-44], [18], [12], [-30], [-40], [-38], [-32], [6], [-38], [-4], [0], [-30], [-28], [-24], [30], [-36], [38], [34], [-10], [-4], [36], [-34], [-10], [48], [24], [30], [4], [0], [0], [-42], [10], [-30], [6], [4], [46], [42], [-20], [22], [24], [12], [-34], [-16], [-42], [10], [4], [40], [-50], [-14], [-36], [-24], [4], [-32], [40], [18], [-22], [46], [-12], [10], [16], [36], [-30], [0], [0], [38], [50], [6], [28], [14], [24], [-38], [32], [-6], [-20], [38], [8], [46], [8], [-36], [-26], [-6], [-24], [-10], [30], [-28], [-22], [-8], [2], [-20], [12], [-66], [12], [58], [18], [62], [18], [-40], [14], [48], [-42], [-30], [28], [-18], [-56], [28], [-6], [4], [-14], [-42], [-56], [12], [0], [-22], [48], [18], [-48], [30], [38], [32], [-46], [-24], [-68], [-10], [42], [48], [-56], [-4], [46], [-20], [40], [42], [20], [-40], [-14], [54], [4], [40], [-60], [20], [24], [-34], [34], [72], [8], [-60], [-20], [64], [-2], [50], [16], [74], [-34], [52], [-42], [-20], [-42], [-22], [80], [60], [-42], [-34], [-14], [-4], [78], [10], [-4], [-74], [-18], [20], [-22], [16], [50], [0], [60], [-18], [74], [44], [-64], [16], [-24], [-42], [-20], [-16], [-14], [22], [-64], [-62], [-18], [12], [-22], [60], [46], [30], [64], [6], [20], [-76], [42], [14], [-64], [-52], [-28], [50], [4], [-18], [0], [-26], [-48], [22], [-62], [-36], [48], [42], [-60], [8], [-46], [18], [-4], [10], [30], [8], [-38], [50], [36], [76], [48], [-42], [-2], [-50], [32], [-60], [-84], [-36], [46], [-14], [10], [32], [22], [42], [70], [40], [-66], [60], [-90], [-20], [72], [6], [-76], [58], [-50], [32], [-10], [10], [56], [-20], [2], [-48], [-26], [-54], [16], [-36], [44], [-54], [-82], [-32], [-38], [12], [4], [-56], [-42], [40], [-34], [36], [-64], [50], [50], [-38], [30], [58], [-16], [50], [-60], [80], [-40], [22], [12], [96], [-30], [6], [28], [-52], [-32], [-70], [8], [74], [52], [86], [46], [-46], [-8], [10], [70], [-40], [-82], [-62], [4], [36], [-14], [6], [-60], [44], [-22], [-50], [-24], [56], [-30], [24], [-2], [-26], [-40], [-108], [-70], [-18], [-4], [10], [-52], [-40]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3120_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3120_2_a_k();". // 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_3120_2_a_k(:prec:=1) chi := MakeCharacter_3120_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_3120_2_a_k();". // 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_3120_2_a_k( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3120_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![0, 1]>,<11,R![4, 1]>,<17,R![-2, 1]>,<19,R![-4, 1]>,<31,R![-8, 1]>],Snew); return Vf; end function;