// Make newform 3528.2.a.v in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3528_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_3528_2_a_v();". // 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_3528_2_a_v();". // 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_3528_a();" function MakeCharacter_3528_a() N := 3528; order := 1; char_gens := [2647, 1765, 785, 1081]; 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_3528_a_Hecke(Kf) return MakeCharacter_3528_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], [2], [0], [0], [-6], [-2], [-4], [4], [10], [8], [6], [-2], [-4], [8], [10], [12], [2], [12], [12], [14], [-8], [12], [-2], [-10], [-6], [0], [0], [-2], [-18], [8], [-4], [-2], [12], [-22], [-8], [18], [-20], [0], [2], [16], [-14], [20], [18], [-6], [-8], [20], [0], [20], [-14], [-2], [-20], [-10], [-20], [6], [-12], [-6], [-8], [22], [-18], [4], [-6], [-4], [0], [22], [18], [-4], [2], [-24], [-14], [30], [12], [16], [-10], [12], [0], [2], [2], [6], [-18], [20], [-26], [-12], [14], [24], [-8], [-34], [10], [18], [-40], [36], [-8], [-8], [24], [4], [40], [42], [6], [-20], [-2], [-20], [2], [4], [30], [20], [-18], [28], [-18], [44], [-10], [-16], [-26], [30], [12], [-16], [-10], [-4], [-48], [10], [-8], [10], [-14], [-6], [24], [-44], [-22], [22], [0], [16], [26], [-20], [-20], [32], [-26], [-18], [-34], [10], [-28], [18], [6], [-36], [-6], [-16], [0], [-38], [0], [26], [-18], [20], [-20], [14], [-2], [4], [-48], [44], [20], [-8], [-26], [54], [42], [56], [-58], [-48], [4], [6], [-24], [0], [26], [-30], [-38], [32], [18], [60], [10], [-48], [-10], [20], [-30], [-16], [50], [-56], [44], [22], [22], [-36], [42], [62], [52], [42], [-48], [22], [48], [-28], [-30], [8], [6], [-62], [14], [22], [64], [34], [0], [-14], [38], [-60], [-30], [16], [-24], [70], [-28], [18], [-30], [40], [36], [32], [6], [48], [14], [-60], [-22], [-42], [-64], [46], [0], [-68], [-10], [-66], [-20], [-64], [-24], [-2], [-20], [-32], [-10], [-28], [-48], [-26], [-6], [8], [-72], [64], [-20], [-8], [46], [6], [-32], [56], [20], [36], [-20], [62], [62], [44], [-18], [-14], [-24], [38], [52], [-54], [-42], [56], [0], [66], [2], [30], [4], [2], [70], [-12], [-30], [-46], [52], [-2], [0], [-42], [-32], [-80], [-2], [-70], [-60], [0], [-64], [72], [-6], [76], [-12], [-78], [-22], [0], [6], [-30], [-20], [-50], [-20], [14], [-14], [-48], [-46], [60], [-44], [-74], [42], [24], [72], [-12], [-46], [-72], [26], [-4], [38], [-16], [-14], [38], [4], [-44], [14], [-20], [44], [86], [6], [28], [-22], [-78], [-32], [-26], [70], [-68], [-76], [36], [18], [-18], [-6], [-40], [52], [20], [-4], [-2], [70], [-2], [8], [54], [-18], [66], [56], [58], [-72], [-46], [28], [-32], [-22], [-20], [58], [-30], [0], [-42], [-42], [-48], [-28], [-18], [-36], [38], [-66], [-12], [0], [-60], [-70], [-54], [0], [10], [64], [52], [-12], [26], [80], [30], [-20], [12], [-50], [62], [14], [-62], [-26], [40], [-18], [4], [72], [-64], [-62], [-76], [-8], [-94], [-62], [0], [12], [-52], [-22], [-64], [22], [-92], [-30], [18], [-2], [72], [-18], [-46], [-56], [-82], [-34], [-60], [84], [30], [-78], [-56], [-92], [-54], [42], [-4], [-16], [-18], [16], [90], [10], [-12], [36], [-42], [18], [32], [30], [-20], [16]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3528_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3528_2_a_v();". // 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_3528_2_a_v(:prec:=1) chi := MakeCharacter_3528_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_3528_2_a_v();". // 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_3528_2_a_v( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3528_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-2, 1]>,<11,R![0, 1]>,<13,R![6, 1]>,<23,R![-4, 1]>],Snew); return Vf; end function;