// Make newform 3192.2.a.f in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3192_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_3192_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_3192_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_3192_a();" function MakeCharacter_3192_a() N := 3192; order := 1; char_gens := [799, 1597, 2129, 913, 1009]; 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_3192_a_Hecke(Kf) return MakeCharacter_3192_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], [0], [1], [-4], [2], [0], [1], [0], [8], [-8], [-10], [2], [4], [-2], [12], [-4], [-10], [-8], [6], [6], [-16], [6], [-6], [2], [-8], [-16], [-6], [2], [-4], [-12], [10], [-6], [0], [-6], [12], [6], [12], [-12], [6], [-18], [-2], [-24], [-14], [-2], [-12], [-12], [0], [16], [18], [-18], [-20], [10], [-6], [6], [-12], [26], [-24], [-22], [12], [-32], [-2], [-28], [14], [-6], [-8], [-8], [22], [-24], [30], [-16], [-8], [4], [-26], [16], [36], [18], [22], [-16], [22], [30], [18], [6], [-14], [32], [16], [-8], [6], [-24], [24], [18], [2], [16], [-8], [-20], [30], [-22], [38], [36], [-2], [28], [2], [-16], [20], [-4], [2], [10], [20], [10], [10], [-16], [2], [-46], [0], [-48], [-24], [-8], [-10], [30], [14], [2], [34], [42], [46], [-20], [-18], [26], [42], [-8], [-22], [20], [6], [20], [46], [-20], [-54], [-26], [-20], [-50], [-6], [28], [-30], [-16], [2], [46], [40], [6], [-30], [-12], [34], [6], [8], [36], [20], [28], [50], [-40], [32], [-38], [42], [-44], [-56], [32], [-28], [-24], [-56], [48], [6], [-26], [48], [26], [10], [28], [-58], [8], [-12], [-20], [-26], [-56], [-18], [-48], [20], [-22], [18], [-40], [24], [-30], [4], [-42], [-38], [-2], [12], [0], [-54], [44], [-18], [-26], [6], [-32], [66], [20], [-48], [-2], [14], [-22], [-32], [20], [10], [-26], [60], [-50], [16], [40], [-24], [24], [14], [8], [22], [-6], [-46], [50], [32], [60], [-40], [-36], [-26], [30], [42], [48], [32], [-26], [-20], [8], [-52], [-52], [-54], [46], [6], [-16], [68], [-34], [52], [28], [-26], [-42], [26], [-36], [-64], [36], [0], [30], [44], [-12], [-74], [-12], [36], [22], [-52], [-30], [-14], [-16], [-78], [74], [-30], [-28], [-20], [52], [4], [36], [-78], [46], [28], [-26], [-56], [2], [52], [0], [46], [-54], [58], [-12], [72], [46], [38], [-36], [60], [2], [-48], [36], [-26], [-38], [22], [-76], [-28], [-58], [60], [-48], [-80], [48], [-40], [-22], [48], [-72], [-78], [12], [-10], [-54], [-14], [-4], [34], [30], [-54], [64], [-48], [48], [58], [-54], [-2], [-26], [18], [-4], [-2], [42], [68], [22], [-6], [-36], [-4], [-2], [54], [46], [40], [60], [-58], [28], [42], [-10], [10], [-74], [-8], [-90], [18], [30], [-24], [-48], [-2], [10], [76], [40], [56], [-68], [-2], [-42], [8], [-50], [70], [-42], [-2], [-90], [-82], [-26], [44], [86], [2], [0], [-62], [-92], [4], [2], [-20], [-60], [28], [-44], [-48], [38], [-44], [-24], [-38], [52], [82], [-30], [-54], [16], [18], [60], [-4], [-16], [50], [44], [-12], [94], [86], [44], [-28], [54], [54], [0], [74], [-40], [50], [50], [50], [32], [78], [74], [28], [-10], [8], [-4], [70], [66], [10], [82], [-4], [106], [-84], [38], [56], [-48], [24], [-16], [-58], [12], [-36], [-58], [78], [-66], [54], [32], [46]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3192_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3192_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_3192_2_a_f(:prec:=1) chi := MakeCharacter_3192_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_3192_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_3192_2_a_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3192_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![0, 1]>,<11,R![4, 1]>,<17,R![0, 1]>],Snew); return Vf; end function;