// Make newform 2310.2.a.s in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2310_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_2310_2_a_s();". // 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_2310_2_a_s();". // 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_2310_a();" function MakeCharacter_2310_a() N := 2310; order := 1; char_gens := [1541, 1387, 661, 211]; 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_2310_a_Hecke(Kf) return MakeCharacter_2310_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 := [[1], [1], [-1], [-1], [1], [4], [0], [-4], [6], [-2], [8], [4], [10], [-8], [8], [-10], [4], [2], [-6], [4], [-2], [4], [4], [-4], [2], [-6], [8], [4], [-12], [16], [-8], [4], [-8], [-8], [2], [4], [-10], [-6], [10], [-14], [-4], [-8], [-24], [-2], [-14], [-16], [-2], [-8], [12], [-12], [18], [-6], [-10], [12], [18], [-4], [6], [24], [2], [-16], [-14], [6], [-22], [-18], [22], [-18], [4], [-22], [28], [10], [18], [-18], [-24], [-34], [-12], [0], [6], [18], [10], [-22], [-16], [30], [-30], [34], [0], [24], [-6], [-10], [-2], [-20], [12], [-16], [20], [4], [16], [42], [-34], [4], [22], [-8], [-28], [-30], [8], [-20], [-6], [-10], [44], [-12], [0], [-38], [-40], [-46], [8], [18], [40], [2], [36], [0], [34], [44], [-48], [-22], [42], [-8], [-30], [6], [-50], [42], [8], [-4], [26], [48], [40], [-36], [-6], [-22], [-26], [-14], [6], [4], [4], [-18], [16], [44], [-16], [-18], [44], [44], [30], [-34], [-22], [-44], [50], [-34], [-2], [12], [-32], [0], [50], [50], [-28], [46], [-40], [-24], [44], [16], [-24], [-40], [-6], [-58], [-24], [32], [22], [-10], [20], [36], [18], [-2], [36], [-6], [12], [-28], [36], [8], [-42], [-6], [-18], [28], [-22], [0], [46], [44], [-8], [-14], [-60], [-66], [-22], [60], [68], [-18], [-62], [8], [62], [10], [44], [30], [56], [36], [-16], [-18], [-2], [-30], [48], [12], [28], [46], [40], [-22], [54], [-22], [4], [-32], [-46], [48], [-48], [36], [-18], [60], [40], [20], [-8], [0], [-60], [-64], [-64], [-54], [18], [-34], [60], [-18], [-16], [36], [32], [-42], [-44], [-48], [8], [-32], [62], [-76], [-22], [-38], [58], [-42], [-18], [16], [38], [-18], [38], [-10], [-24], [52], [-14], [52], [-42], [46], [-10], [-72], [20], [-26], [56], [-6], [50], [80], [14], [-72], [68], [-16], [50], [-36], [74], [8], [-18], [10], [-38], [20], [-30], [-18], [-20], [14], [82], [-60], [42], [12], [58], [-66], [-36], [10], [36], [38], [-34], [-46], [52], [-32], [-40], [18], [-12], [-84], [-4], [6], [-26], [-86], [-48], [-26], [4], [86], [-80], [-86], [10], [-56], [-60], [54], [-66], [-44], [38], [-30], [24], [-36], [-24], [-66], [-72], [-30], [88], [24], [78], [-60], [-90], [-44], [-82], [28], [-16], [48], [42], [-24], [-66], [-32], [52], [-50], [-12], [54], [-36], [-38], [94], [-20], [24], [-4], [-22], [-68], [38], [-86], [-54], [78], [50], [24], [-28], [-30], [-10], [-72], [-34], [68], [-18], [-16], [78], [-80], [88], [-56], [2], [-70], [-34], [-78], [10], [56], [-88], [-70], [-40], [-80], [88], [46], [-64], [36], [-82], [-54], [12], [-32], [60], [14], [8], [-36], [-36], [50], [-26], [4], [-64], [-6], [70], [8], [80], [76], [92], [-52], [34], [86], [24], [98], [50], [-98], [-46], [-8], [22], [-6], [-58], [4], [-94], [44], [6], [46], [56], [4], [-30], [36]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2310_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2310_2_a_s();". // 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_2310_2_a_s(:prec:=1) chi := MakeCharacter_2310_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_2310_2_a_s();". // 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_2310_2_a_s( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2310_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<13,R![-4, 1]>,<17,R![0, 1]>,<19,R![4, 1]>,<23,R![-6, 1]>,<29,R![2, 1]>,<31,R![-8, 1]>],Snew); return Vf; end function;