// Make newform 7440.2.a.r in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_7440_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_7440_2_a_r();". // 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_7440_2_a_r();". // 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_7440_a();" function MakeCharacter_7440_a() N := 7440; order := 1; char_gens := [6511, 1861, 4961, 2977, 5521]; 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_7440_a_Hecke(Kf) return MakeCharacter_7440_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], [0], [6], [-2], [-4], [0], [-2], [-8], [-1], [-6], [-2], [-4], [-4], [-6], [0], [4], [4], [8], [-4], [4], [0], [-2], [14], [2], [-16], [-12], [-10], [6], [-14], [0], [8], [2], [-14], [-12], [-2], [4], [10], [14], [2], [8], [8], [-18], [-18], [20], [4], [-18], [-12], [16], [6], [24], [18], [-18], [-26], [-18], [-16], [-16], [18], [-26], [12], [-30], [-4], [24], [-12], [-26], [18], [-8], [-16], [-30], [0], [-24], [10], [18], [24], [-6], [-20], [38], [-26], [14], [0], [26], [-40], [-28], [0], [-12], [14], [-8], [0], [14], [36], [-24], [34], [34], [38], [16], [-36], [6], [-20], [-2], [44], [38], [-44], [-2], [38], [-38], [-32], [34], [-48], [30], [28], [-22], [-6], [26], [-8], [-22], [44], [18], [34], [16], [-26], [32], [6], [4], [-44], [30], [-4], [-20], [12], [6], [2], [22], [8], [2], [-18], [30], [26], [8], [2], [-46], [-24], [16], [-6], [-32], [-32], [0], [10], [-18], [22], [-22], [-50], [38], [-20], [4], [-44], [16], [-16], [46], [14], [52], [12], [8], [-6], [-36], [26], [-10], [-20], [18], [-46], [-46], [-30], [32], [-16], [-42], [-24], [30], [20], [-26], [44], [0], [12], [-22], [26], [-56], [-16], [36], [-46], [4], [-18], [-48], [20], [36], [14], [48], [28], [-60], [22], [62], [-54], [12], [26], [40], [26], [58], [60], [-10], [-16], [-12], [34], [36], [-36], [12], [-28], [44], [4], [10], [8], [34], [-14], [-46], [-60], [-56], [-14], [16], [52], [20], [-62], [32], [-42], [28], [46], [20], [24], [54], [40], [-38], [30], [-42], [-46], [72], [-12], [46], [-26], [-20], [-20], [-48], [70], [62], [46], [-16], [-14], [66], [-62], [-26], [30], [60], [38], [-68], [22], [-42], [36], [-44], [-56], [2], [-28], [-20], [-30], [-58], [44], [-10], [82], [-80], [20], [12], [62], [28], [68], [76], [-66], [-78], [4], [0], [-56], [-10], [-28], [0], [68], [-42], [-8], [42], [-2], [-4], [-24], [84], [26], [68], [48], [-26], [34], [0], [26], [22], [8], [12], [-30], [-34], [-68], [22], [-84], [-54], [14], [-52], [10], [-48], [-84], [-82], [-6], [-84], [6], [18], [70], [-12], [2], [76], [6], [10], [-80], [-4], [-78], [50], [-50], [74], [16], [60], [52], [60], [52], [54], [-26], [58], [-10], [16], [-72], [4], [-66], [-24], [-42], [-80], [24], [-6], [-14], [8], [30], [22], [-18], [-4], [20], [-18], [-36], [40], [26], [74], [-2], [24], [12], [48], [-14], [14], [-98], [24], [56], [-44], [-70], [-24], [58], [58], [-48], [-38], [66], [12], [48], [-64], [-42], [80], [54], [4], [0], [-30], [64], [-6], [-22], [22], [56], [-60], [40], [74], [24], [-10], [62], [44], [56], [-18], [-56], [98], [84], [-64], [82], [30], [-80], [-62], [-44], [-54], [44], [-94], [-22], [62], [12], [16], [-6], [0], [-12], [70], [-22], [-16], [-46], [62], [-36], [-6], [42], [-60], [-74], [8], [50], [64], [-86], [-70], [-2], [52], [-52]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_7440_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_7440_2_a_r();". // 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_7440_2_a_r(:prec:=1) chi := MakeCharacter_7440_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(3067) - 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_7440_2_a_r();". // 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_7440_2_a_r( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7440_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![0, 1]>,<11,R![-6, 1]>,<13,R![2, 1]>,<17,R![4, 1]>,<19,R![0, 1]>],Snew); return Vf; end function;