// Make newform 2280.2.a.g in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2280_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_2280_2_a_g();". // 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_2280_2_a_g();". // 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_2280_a();" function MakeCharacter_2280_a() N := 2280; order := 1; char_gens := [1711, 1141, 761, 457, 1921]; 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_2280_a_Hecke(Kf) return MakeCharacter_2280_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], [1], [-2], [-2], [-4], [-2], [-1], [0], [-4], [8], [-8], [-12], [-6], [0], [-2], [0], [2], [-8], [8], [6], [8], [-8], [-4], [-12], [10], [-4], [12], [-14], [6], [0], [-6], [2], [0], [10], [0], [-14], [6], [-8], [26], [0], [10], [10], [16], [-22], [4], [-4], [-20], [-20], [-6], [-6], [6], [-22], [22], [6], [28], [20], [4], [-22], [-16], [14], [-6], [16], [-14], [-14], [14], [4], [-20], [-16], [-26], [18], [34], [-6], [-36], [4], [8], [30], [2], [-8], [22], [-18], [14], [4], [24], [0], [-12], [4], [-22], [2], [34], [12], [18], [4], [-14], [24], [-28], [40], [0], [-28], [14], [-32], [-18], [28], [-12], [40], [18], [-4], [34], [28], [30], [-36], [10], [22], [40], [0], [-4], [-34], [12], [-14], [-20], [-22], [28], [-42], [-44], [-48], [2], [-22], [6], [10], [-42], [-44], [-40], [32], [22], [30], [-14], [30], [-52], [-6], [-30], [4], [18], [2], [-36], [2], [24], [38], [2], [-28], [-16], [-44], [6], [46], [-8], [-20], [0], [32], [42], [-34], [60], [-24], [-10], [-6], [-28], [-14], [-40], [40], [-14], [-6], [14], [12], [-42], [-38], [10], [40], [10], [-8], [18], [-12], [-50], [-34], [-52], [52], [-18], [-40], [-30], [-40], [-44], [6], [30], [-16], [28], [-44], [-60], [-12], [-18], [-2], [6], [-2], [0], [-20], [32], [-20], [-70], [-22], [-30], [-24], [-52], [6], [12], [18], [-34], [-22], [36], [-60], [-42], [26], [0], [48], [-54], [62], [-8], [-68], [22], [60], [-10], [6], [-60], [56], [-62], [-2], [20], [-16], [-36], [26], [-32], [30], [6], [-30], [-28], [12], [68], [-26], [10], [-30], [-18], [6], [24], [-20], [12], [-2], [-14], [36], [-26], [42], [-34], [10], [-68], [-58], [-10], [-44], [-36], [6], [-52], [46], [-44], [-28], [-14], [52], [34], [38], [-16], [-18], [44], [76], [74], [56], [-30], [70], [82], [24], [-40], [48], [-50], [-22], [-46], [26], [22], [-32], [12], [2], [0], [-54], [60], [-44], [-18], [-72], [10], [-80], [-30], [46], [-42], [24], [4], [40], [68], [60], [-10], [50], [6], [56], [-26], [-24], [-64], [60], [-70], [2], [-72], [34], [6], [68], [70], [84], [8], [-58], [42], [-36], [-12], [32], [90], [-18], [-42], [-44], [8], [-40], [48], [-10], [66], [30], [-10], [-84], [-54], [80], [72], [-2], [28], [2], [-8], [-28], [-2], [-20], [0], [6], [-20], [10], [18], [-2], [30], [-18], [-72], [-6], [6], [48], [-20], [38], [-24], [-18], [-52], [70], [26], [-52], [36], [72], [40], [58], [-96], [-50], [-22], [82], [44], [-96], [-26], [-14], [-30], [60], [32], [-12], [34], [-26], [24], [-10], [-58], [14], [6], [-60], [0], [-56], [-80], [100], [-82], [-34], [94], [-40], [6], [12], [-64], [-22], [44], [-64], [-10], [-4], [58], [52], [-72], [-58], [-102], [12], [48], [74], [-64], [-56], [84], [64], [56], [40], [-86], [84], [18], [88], [94]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2280_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2280_2_a_g();". // 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_2280_2_a_g(:prec:=1) chi := MakeCharacter_2280_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_2280_2_a_g();". // 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_2280_2_a_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2280_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]>],Snew); return Vf; end function;