// Make newform 9408.2.a.g in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_9408_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_9408_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_9408_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_9408_a();" function MakeCharacter_9408_a() N := 9408; order := 1; char_gens := [1471, 6469, 3137, 4609]; 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_9408_a_Hecke(Kf) return MakeCharacter_9408_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], [-2], [0], [-4], [-6], [2], [-4], [4], [2], [8], [10], [2], [8], [0], [10], [12], [10], [-8], [-12], [-2], [0], [-12], [-6], [-2], [-10], [-8], [-12], [2], [-14], [-8], [20], [18], [-20], [-6], [16], [2], [16], [0], [6], [-20], [2], [-12], [-14], [-6], [16], [-16], [-8], [-20], [18], [-14], [-12], [30], [4], [18], [12], [-2], [-32], [18], [-30], [12], [6], [-4], [8], [14], [2], [-8], [-14], [-28], [18], [-6], [12], [-24], [34], [-8], [-8], [-38], [34], [10], [22], [-12], [34], [-36], [6], [8], [28], [18], [-6], [-26], [-16], [-36], [0], [32], [-36], [-16], [0], [-18], [-38], [-28], [26], [-32], [-30], [20], [18], [16], [-2], [-12], [42], [36], [-10], [8], [10], [18], [20], [0], [26], [-28], [-16], [-14], [44], [18], [50], [-18], [28], [-12], [-30], [-38], [8], [-40], [-14], [32], [28], [-16], [26], [-30], [-26], [-42], [4], [-42], [10], [4], [10], [-32], [4], [-14], [-16], [42], [18], [-12], [-52], [-14], [42], [16], [-16], [48], [12], [-8], [-22], [-10], [6], [44], [-54], [-8], [-28], [-22], [24], [-8], [-62], [2], [30], [36], [-30], [20], [-38], [32], [2], [40], [-6], [32], [-6], [-8], [12], [-38], [2], [20], [30], [10], [4], [-6], [-8], [22], [-12], [0], [-42], [-44], [-14], [-14], [2], [2], [-56], [-30], [0], [10], [-10], [-12], [-66], [64], [-36], [-6], [36], [-62], [-2], [-16], [-12], [-40], [46], [-56], [-46], [-20], [34], [-6], [32], [66], [16], [-28], [26], [-6], [-12], [-32], [-60], [-14], [-28], [24], [-54], [-4], [0], [-34], [26], [12], [16], [-28], [-60], [-72], [26], [18], [48], [-48], [-52], [-16], [-36], [-22], [-38], [-28], [62], [-18], [-12], [26], [-28], [6], [14], [40], [-12], [10], [-22], [-78], [28], [34], [-30], [-8], [-38], [-14], [-72], [-34], [-80], [-26], [-64], [12], [26], [-6], [28], [-48], [80], [56], [2], [52], [44], [34], [-6], [40], [2], [66], [-60], [34], [60], [26], [-74], [24], [-58], [20], [-36], [46], [18], [-32], [60], [-32], [-14], [36], [-30], [76], [-46], [0], [-54], [-54], [-40], [-4], [30], [12], [-20], [6], [10], [36], [26], [-50], [-24], [-54], [14], [16], [28], [12], [-38], [10], [-62], [40], [84], [56], [28], [-14], [-30], [-66], [-8], [-78], [66], [-42], [-8], [-30], [60], [34], [-88], [32], [54], [-60], [58], [50], [0], [-46], [2], [-24], [-44], [-30], [44], [-14], [-70], [-4], [-60], [-12], [90], [46], [56], [-70], [-84], [20], [-60], [-22], [80], [-38], [-76], [36], [14], [82], [-42], [70], [74], [72], [18], [-68], [48], [56], [-22], [40], [-48], [18], [62], [-12], [68], [60], [26], [0], [42], [-8], [42], [26], [18], [-64], [-78], [-74], [0], [-38], [-14], [-20], [20], [-34], [10], [-44], [-80], [-38], [-66], [12], [8], [-62], [-24], [2], [-78], [36], [12], [-82], [94], [4], [18], [12], [-16], [6], [-36], [0], [-16], [10], [42], [-102], [42], [48], [32], [4], [-38], [42], [-28], [70], [-78], [-28], [-88], [102], [50], [-16], [56], [-20], [10], [50], [-6], [82], [-12], [-70], [-70], [-40], [24], [4], [-62], [92], [-94], [-64], [44], [74], [60], [-104], [-28], [-24], [50], [52], [42], [66], [64], [32], [90], [-26], [-94], [-98], [6], [-104], [44], [66], [44], [92], [-88], [58], [56], [-54], [14], [84], [58], [-28], [58], [-8], [-112], [-30]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_9408_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_9408_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_9408_2_a_g(:prec:=1) chi := MakeCharacter_9408_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(3581) - 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_9408_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_9408_2_a_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_9408_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![2, 1]>,<11,R![4, 1]>,<13,R![6, 1]>,<17,R![-2, 1]>,<19,R![4, 1]>,<31,R![-8, 1]>],Snew); return Vf; end function;