// Make newform 1260.2.a.f in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1260_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_1260_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_1260_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_1260_a();" function MakeCharacter_1260_a() N := 1260; order := 1; char_gens := [631, 281, 757, 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_1260_a_Hecke(Kf) return MakeCharacter_1260_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], [1], [-1], [-4], [0], [-2], [-6], [-6], [0], [2], [2], [-2], [4], [-8], [-10], [-4], [-2], [12], [-8], [8], [-8], [4], [-10], [-4], [18], [4], [-6], [-18], [10], [-8], [-20], [-6], [-2], [0], [8], [-8], [20], [12], [22], [-24], [22], [12], [2], [10], [-10], [-8], [4], [16], [-14], [-18], [20], [10], [12], [-2], [-26], [30], [6], [-6], [16], [-4], [26], [16], [0], [32], [14], [-32], [10], [-22], [-26], [18], [4], [28], [14], [-12], [-24], [4], [-20], [-28], [-18], [-12], [14], [-24], [-40], [-2], [26], [-24], [-26], [26], [32], [-32], [0], [-16], [24], [-12], [0], [-10], [-14], [4], [30], [12], [-34], [4], [-36], [0], [-8], [28], [-30], [-36], [30], [4], [-6], [-2], [-10], [-40], [-32], [-32], [-28], [-26], [8], [-2], [-26], [42], [-14], [-38], [-16], [10], [0], [24], [16], [16], [-6], [-52], [-50], [6], [22], [6], [28], [-42], [-12], [38], [-12], [-32], [34], [14], [40], [-52], [-30], [6], [50], [2], [-38], [36], [-8], [20], [-12], [20], [14], [-28], [46], [22], [-54], [-8], [36], [6], [-20], [4], [-52], [-2], [-22], [28], [50], [24], [-54], [50], [-22], [-20], [12], [-64], [-38], [8], [12], [38], [-2], [54], [-6], [34], [-52], [22], [0], [-36], [26], [-20], [58], [54], [42], [10], [46], [-18], [52], [44], [-10], [12], [66], [-12], [-26], [26], [-6], [-60], [-58], [-66], [6], [-8], [28], [-48], [-34], [64], [-30], [38], [30], [-70], [50], [-36], [-24], [-32], [14], [38], [16], [28], [16], [-50], [-30], [-56], [60], [-56], [-12], [70], [-14], [64], [8], [-34], [38], [-24], [-50], [-54], [0], [8], [-36], [44], [-14], [-38], [-30], [-18], [54], [-66], [68], [50], [28], [-42], [-72], [-24], [-70], [10], [-40], [66], [2], [-36], [-62], [4], [54], [-6], [-60], [-76], [36], [68], [80], [10], [26], [-42], [-76], [-76], [-16], [-36], [42], [20], [60], [-74], [6], [-46], [6], [60], [-68], [-34], [-68], [30], [-6], [-38], [-10], [-12], [44], [36], [22], [52], [54], [-40], [18], [-26], [-54], [-48], [38], [84], [-24], [20], [44], [-30], [50], [-60], [48], [-36], [76], [-2], [-50], [-14], [48], [6], [2], [-40], [-12], [38], [42], [-10], [54], [-50], [-60], [68], [-12], [-26], [78], [-14], [-88], [22], [-58], [-18], [8], [6], [-40], [66], [-20], [72], [42], [-70], [54], [12], [32], [-78], [-6], [-24], [68], [18], [-78], [14], [70], [58], [-16], [52], [46], [14], [-8], [22], [-48], [-2], [50], [24], [38], [-38], [4], [-24], [-64], [42], [48], [74], [98], [-96], [54], [30], [0], [36], [32], [44], [-40], [-82], [-46], [80], [48], [8], [10], [-86], [-70], [-48], [-24], [-30], [66], [88], [-6], [38], [-14], [26], [-48], [0], [12], [-28], [-30], [-6], [-56], [50], [38], [60], [24], [102], [-80], [-4], [20], [-54], [100], [-24], [50], [-50], [36], [10], [-80]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1260_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1260_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_1260_2_a_f(:prec:=1) chi := MakeCharacter_1260_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_1260_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_1260_2_a_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1260_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![4, 1]>,<17,R![2, 1]>],Snew); return Vf; end function;