// Make newform 7350.2.a.bd in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_7350_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_7350_2_a_bd();". // 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_7350_2_a_bd();". // 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_7350_a();" function MakeCharacter_7350_a() N := 7350; order := 1; char_gens := [4901, 1177, 2551]; v := [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_7350_a_Hecke(Kf) return MakeCharacter_7350_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], [0], [0], [0], [2], [-6], [4], [0], [-6], [4], [-2], [-6], [-8], [-12], [-6], [12], [-2], [-8], [0], [14], [-16], [12], [-6], [14], [6], [-16], [-12], [14], [-18], [-8], [-12], [6], [4], [18], [-16], [2], [16], [12], [-6], [0], [-2], [0], [22], [18], [4], [-28], [-16], [-12], [-2], [6], [0], [-26], [-12], [-6], [0], [-18], [-20], [22], [-30], [20], [-6], [20], [0], [-10], [-30], [-4], [22], [-12], [-26], [18], [0], [8], [-26], [20], [-36], [-30], [2], [18], [22], [12], [-10], [24], [-34], [28], [-12], [-6], [-2], [30], [16], [12], [24], [16], [24], [-4], [12], [-18], [-6], [20], [38], [-8], [-30], [-36], [-6], [-4], [-34], [-12], [42], [-24], [-2], [32], [-2], [-42], [28], [-16], [18], [-4], [-12], [42], [0], [-26], [-50], [-30], [-36], [-20], [-6], [14], [0], [8], [2], [44], [24], [-40], [22], [-30], [-50], [-6], [-28], [-6], [-30], [4], [-30], [-8], [12], [46], [0], [26], [18], [4], [48], [-50], [42], [40], [-12], [16], [0], [-16], [-54], [-34], [-18], [36], [6], [-56], [-12], [-18], [-36], [56], [26], [2], [-30], [0], [22], [24], [46], [-20], [-30], [-28], [-30], [-40], [46], [-8], [36], [22], [18], [24], [-42], [-26], [44], [-22], [-48], [14], [36], [-52], [30], [36], [18], [2], [-26], [42], [36], [66], [-20], [-22], [46], [36], [18], [28], [-12], [-54], [4], [-2], [54], [40], [-12], [-48], [22], [-56], [18], [-48], [42], [14], [4], [42], [-56], [12], [-34], [18], [48], [8], [24], [22], [52], [32], [42], [44], [12], [-2], [18], [72], [-48], [-36], [-20], [56], [-10], [66], [-24], [-16], [-12], [44], [48], [-50], [18], [24], [-2], [18], [48], [38], [68], [-54], [38], [16], [12], [70], [2], [18], [-68], [-78], [42], [40], [18], [22], [-56], [-58], [-64], [62], [32], [36], [-10], [26], [36], [-60], [56], [-60], [-74], [20], [24], [-26], [-54], [-68], [42], [-30], [-12], [54], [60], [-26], [-18], [28], [18], [36], [20], [14], [42], [80], [36], [-4], [-50], [36], [-2], [0], [-26], [-12], [66], [66], [88], [-72], [70], [-36], [-24], [14], [-54], [-20], [22], [-66], [64], [6], [-74], [20], [44], [-48], [-6], [-34], [-54], [-44], [36], [20], [-36], [-58], [-30], [22], [56], [-26], [-42], [6], [8], [-54], [72], [-2], [40], [-24], [42], [76], [-2], [90], [-64], [62], [42], [48], [36], [30], [48], [22], [-30], [72], [-24], [-76], [46], [90], [-32], [-22], [24], [-92], [24], [42], [52], [-2], [-60], [-72], [14], [-30], [14], [-18], [-90], [16], [30], [76], [84], [-64], [-22], [-80], [12], [50], [-6], [-72], [-28], [48], [-26], [-92], [18], [-28], [-6], [-50], [-66], [-8], [42], [6], [-68], [-50], [42], [44], [36], [38], [-6], [-12], [-4], [-50], [78], [48], [-16], [-78], [36], [-6], [-70], [48], [84], [38], [-30], [-60], [-78], [-20], [0], [22], [-48], [20], [12], [2], [66], [-70], [86], [40], [100], [36], [-54], [-82], [0], [22], [-18], [20], [84], [-2], [94], [40], [72], [-36], [90], [46], [-54], [-34], [12], [-22], [-42], [-4], [32], [-96], [-106], [20], [22], [56], [12], [-30], [-44], [-104], [-12], [48], [50]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_7350_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_7350_2_a_bd();". // 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_7350_2_a_bd(:prec:=1) chi := MakeCharacter_7350_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(3361) - 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_7350_2_a_bd();". // 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_7350_2_a_bd( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7350_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![0, 1]>,<13,R![-2, 1]>,<17,R![6, 1]>,<19,R![-4, 1]>,<23,R![0, 1]>,<31,R![-4, 1]>],Snew); return Vf; end function;