// Make newform 3600.2.a.bh in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3600_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_3600_2_a_bh();". // 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_3600_2_a_bh();". // 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_3600_a();" function MakeCharacter_3600_a() N := 3600; order := 1; char_gens := [3151, 901, 2801, 577]; 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_3600_a_Hecke(Kf) return MakeCharacter_3600_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], [0], [2], [2], [-4], [-2], [-4], [-8], [10], [-4], [0], [0], [-8], [-8], [6], [-14], [-14], [-4], [12], [-6], [12], [-4], [12], [14], [-6], [-14], [12], [2], [18], [18], [-18], [2], [-12], [-10], [-8], [-4], [4], [0], [-2], [-14], [2], [4], [-22], [2], [-8], [-20], [-22], [0], [-2], [-18], [12], [-10], [6], [18], [0], [2], [0], [16], [28], [-4], [2], [-32], [-20], [10], [-30], [12], [2], [-16], [-6], [-26], [24], [26], [20], [-28], [24], [-18], [8], [12], [10], [30], [30], [16], [10], [32], [-24], [-12], [-10], [30], [30], [20], [-4], [22], [-18], [-20], [24], [-14], [-20], [28], [10], [8], [-26], [-8], [20], [-4], [-26], [-36], [42], [-24], [22], [6], [48], [6], [-44], [20], [-24], [12], [24], [-14], [6], [18], [-30], [26], [12], [28], [-22], [14], [-40], [38], [24], [-12], [-24], [36], [-32], [36], [-50], [42], [-28], [30], [24], [28], [-14], [-2], [24], [-38], [-32], [16], [-18], [4], [-48], [20], [-36], [44], [-56], [24], [-16], [-20], [4], [-10], [22], [-56], [-54], [18], [-2], [-2], [24], [-8], [-36], [38], [22], [-14], [-26], [40], [34], [-20], [60], [-52], [-34], [-2], [22], [38], [6], [16], [50], [-16], [-10], [-44], [20], [-10], [44], [-14], [-24], [-20], [14], [28], [58], [-30], [-36], [-38], [-16], [-18], [44], [4], [-18], [-30], [-38], [-4], [40], [-36], [28], [2], [38], [-22], [24], [44], [18], [10], [-52], [24], [-10], [38], [-16], [40], [-66], [12], [22], [-38], [40], [-10], [10], [-52], [-36], [16], [-44], [-60], [0], [-30], [-6], [2], [12], [64], [20], [46], [-38], [-74], [-20], [10], [-38], [-4], [8], [-24], [64], [24], [50], [46], [6], [50], [-12], [54], [-54], [-14], [32], [-42], [-56], [-14], [-28], [-26], [24], [-36], [-70], [-10], [16], [18], [4], [-14], [26], [-32], [2], [-22], [-42], [24], [0], [-40], [-2], [64], [44], [-74], [-54], [24], [0], [-50], [36], [-42], [18], [-64], [6], [28], [-42], [74], [48], [66], [18], [-80], [20], [-84], [62], [12], [70], [12], [52], [16], [42], [60], [-56], [-88], [-26], [26], [84], [-46], [40], [28], [-38], [66], [-50], [62], [-42], [20], [-16], [32], [6], [-82], [38], [-52], [-92], [44], [-36], [-82], [-18], [10], [-78], [-20], [30], [34], [-32], [-34], [-22], [46], [16], [-12], [54], [-52], [-42], [-62], [-22], [2], [38], [-48], [2], [-70], [-64], [-20], [-32], [-16], [66], [12], [14], [-18], [-70], [38], [90], [4], [-40], [-6], [0], [-56], [-54], [-56], [22], [-12], [-2], [-82], [74], [-6], [-66], [84], [-32], [20], [-48], [24], [64], [46], [78], [-54], [76], [16], [-2], [-80], [-24], [52], [-66], [-90], [-6], [-38], [66], [-30], [-12], [32], [60], [4], [-18], [-106], [-102], [-28], [-28], [50], [90], [-88], [-62], [66], [96], [54], [-52], [-32], [14], [66], [42], [-84], [48], [100], [16]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3600_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3600_2_a_bh();". // 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_3600_2_a_bh(:prec:=1) chi := MakeCharacter_3600_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_3600_2_a_bh();". // 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_3600_2_a_bh( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3600_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![-2, 1]>,<11,R![-2, 1]>,<13,R![4, 1]>,<17,R![2, 1]>],Snew); return Vf; end function;