// Make newform 1911.2.a.f in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1911_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_1911_2_a_f();". // 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_1911_2_a_f();". // 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_1911_a();" function MakeCharacter_1911_a() N := 1911; order := 1; char_gens := [638, 1522, 1471]; v := [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_1911_a_Hecke(Kf) return MakeCharacter_1911_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], [1], [-2], [0], [4], [-1], [-2], [0], [0], [-10], [-4], [-2], [-6], [-12], [0], [6], [-12], [2], [-8], [0], [-2], [8], [-4], [2], [-10], [18], [0], [12], [-2], [-6], [-16], [-4], [6], [-12], [-6], [4], [18], [8], [8], [-6], [4], [10], [8], [18], [18], [-8], [-20], [-4], [20], [10], [-14], [-24], [-10], [12], [-26], [24], [-22], [12], [-10], [-10], [-12], [6], [16], [0], [6], [26], [-16], [18], [-12], [26], [2], [24], [-16], [-26], [-24], [16], [22], [-38], [22], [-34], [-4], [-10], [0], [-34], [-32], [-4], [22], [2], [38], [4], [-12], [24], [12], [-12], [-24], [8], [-10], [-18], [-44], [30], [4], [18], [12], [34], [-4], [46], [-28], [26], [-40], [38], [16], [-2], [22], [-24], [20], [2], [40], [8], [6], [28], [-30], [-14], [-22], [-12], [24], [-34], [-26], [24], [-8], [-30], [32], [48], [8], [22], [10], [30], [-10], [32], [-46], [-30], [-8], [-22], [0], [4], [34], [-48], [-30], [46], [44], [16], [-10], [-58], [-44], [48], [12], [-40], [-48], [-30], [-26], [14], [-60], [-30], [-52], [60], [-42], [-16], [16], [26], [-6], [18], [-12], [-22], [-16], [18], [-24], [-42], [-24], [-6], [-16], [-30], [20], [12], [-58], [-30], [-16], [42], [-2], [16], [-46], [-48], [-34], [28], [52], [-50], [36], [30], [34], [-2], [-38], [-48], [-54], [16], [-22], [-2], [36], [34], [-52], [4], [-34], [-28], [2], [-54], [-16], [44], [-40], [22], [8], [-50], [0], [18], [38], [20], [22], [52], [-12], [-26], [54], [32], [-40], [44], [-18], [-20], [44], [42], [68], [-24], [70], [10], [-52], [-16], [4], [-28], [16], [6], [-54], [48], [-4], [-36], [56], [-24], [62], [-62], [-48], [-10], [-6], [-20], [-10], [40], [-6], [38], [-48], [28], [10], [2], [42], [-28], [-22], [42], [56], [-2], [-62], [24], [14], [56], [30], [-4], [-44], [22], [-38], [-60], [-24], [-4], [-16], [82], [40], [16], [34], [-6], [-20], [-74], [-18], [84], [-66], [36], [-2], [50], [0], [26], [28], [-32], [22], [58], [-8], [-4], [-4], [-6], [44], [-46], [48], [6], [16], [66], [10], [-20], [8], [-26], [36], [16], [-58], [-78], [-44], [-86], [18], [-36], [-66], [30], [-40], [80], [64], [30], [14], [-90], [-40], [-92], [40], [-28], [-74], [-14], [-2], [8], [30], [-62], [-26], [80], [26], [36], [26], [-32], [-80], [-22], [72], [-30], [66], [16], [-58], [-2], [-80], [12], [-54], [-56], [78], [54], [-48], [84], [20], [-38], [6], [-92], [-54], [84], [60], [64], [-42], [0], [30], [-36], [48], [-82], [62], [54], [30], [-42], [-4], [54], [-40], [-16], [60], [-54], [56], [16], [-22], [22], [68], [52], [-40], [10], [4], [-90], [52], [-30], [58], [-30], [-92], [-30], [-50], [-32], [46], [-10], [0], [-12], [-50], [38], [-12], [-68], [-6], [-30], [-40], [-32], [-6], [64], [-26], [-78], [8], [4], [70], [-34], [-36], [-66], [16], [-48]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1911_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1911_2_a_f();". // 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_1911_2_a_f(:prec:=1) chi := MakeCharacter_1911_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_1911_2_a_f();". // 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_1911_2_a_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1911_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-1, 1]>,<5,R![2, 1]>],Snew); return Vf; end function;