// Make newform 3600.2.a.k 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_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_3600_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_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], [-4], [4], [0], [4], [-2], [-2], [0], [4], [-2], [6], [-6], [4], [-12], [-10], [-14], [8], [8], [-16], [2], [-6], [16], [-6], [-14], [-10], [-6], [-16], [6], [-12], [-8], [-4], [-18], [8], [-4], [-2], [18], [12], [-4], [22], [0], [-16], [12], [8], [-4], [-6], [-2], [-6], [24], [-16], [-22], [-20], [0], [30], [6], [16], [-12], [-18], [6], [-28], [2], [24], [8], [-12], [-28], [-16], [22], [2], [0], [-8], [-10], [-20], [-20], [6], [30], [-36], [14], [14], [12], [-2], [16], [0], [24], [-6], [-6], [-24], [18], [26], [-18], [32], [-2], [-4], [4], [-34], [-34], [-10], [-10], [-2], [34], [-12], [18], [18], [20], [32], [38], [16], [40], [18], [-10], [-20], [8], [-4], [40], [6], [-18], [2], [-36], [-36], [14], [-16], [12], [-38], [-36], [-22], [-38], [16], [-18], [36], [-12], [30], [16], [52], [38], [-2], [20], [2], [-12], [42], [36], [18], [2], [-42], [10], [24], [28], [8], [44], [-42], [44], [-10], [-50], [18], [-22], [48], [24], [-38], [-40], [-14], [30], [24], [-50], [-36], [-32], [46], [32], [-12], [14], [36], [20], [30], [-24], [40], [16], [26], [-12], [58], [-14], [-6], [38], [-12], [12], [-24], [6], [-10], [-4], [14], [-2], [0], [32], [-54], [-4], [-54], [-18], [-8], [10], [-44], [64], [-50], [-18], [-16], [-28], [-2], [4], [4], [32], [50], [10], [-28], [-32], [18], [-46], [22], [-40], [-62], [22], [-50], [34], [-20], [-34], [-40], [-30], [-54], [-2], [50], [-24], [-64], [46], [60], [4], [-60], [0], [-2], [-26], [42], [-10], [-60], [20], [72], [2], [-44], [34], [-46], [48], [-56], [38], [-12], [60], [-26], [12], [30], [-30], [-2], [-52], [60], [-10], [10], [-4], [24], [74], [62], [-38], [-28], [-48], [-12], [-50], [70], [6], [36], [46], [82], [-8], [-64], [-48], [-46], [38], [-30], [66], [-28], [-10], [-40], [-62], [-26], [-70], [-48], [32], [-36], [-40], [-70], [26], [-2], [-56], [-36], [-12], [-42], [-32], [-12], [-76], [2], [56], [-44], [16], [-30], [52], [48], [-42], [-14], [-24], [28], [54], [54], [-34], [-2], [-14], [54], [-36], [32], [0], [-86], [-4], [88], [-86], [42], [8], [-22], [20], [-90], [10], [-44], [-26], [68], [-64], [-14], [36], [-26], [10], [64], [-30], [-90], [-52], [40], [62], [-40], [60], [12], [-2], [-6], [48], [92], [44], [8], [90], [42], [74], [8], [64], [-4], [64], [14], [-44], [-42], [74], [-12], [-14], [-56], [36], [34], [50], [-12], [-100], [6], [46], [-88], [76], [-4], [-96], [-32], [-22], [88], [-30], [40], [-2], [-32], [52], [78], [-48], [20], [-26], [-54], [-90], [36], [4], [-14], [-40], [-88], [32], [42], [36], [50], [-54], [-80], [-42], [104], [-66], [88], [28], [46], [-34], [76], [80], [12], [-6], [-20], [56], [50], [-96], [-50], [64], [-82], [-74], [52], [-102], [-12], [24], [84], [34], [82], [52], [8]]; 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_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_3600_2_a_k(: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_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_3600_2_a_k( : 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![4, 1]>,<13,R![-4, 1]>,<17,R![0, 1]>],Snew); return Vf; end function;