// Make newform 7056.2.a.bf in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_7056_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_7056_2_a_bf();". // 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_7056_2_a_bf();". // 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_7056_a();" function MakeCharacter_7056_a() N := 7056; order := 1; char_gens := [6175, 1765, 785, 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_7056_a_Hecke(Kf) return MakeCharacter_7056_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], [0], [0], [0], [0], [7], [0], [-7], [0], [0], [-7], [-1], [0], [-5], [0], [0], [0], [-14], [-11], [0], [7], [13], [0], [0], [-14], [0], [-7], [0], [17], [0], [19], [0], [0], [-7], [0], [4], [-14], [-8], [0], [0], [0], [7], [0], [-25], [0], [-28], [16], [-28], [0], [7], [0], [0], [-14], [0], [0], [0], [0], [-28], [-31], [0], [-7], [0], [35], [0], [-35], [0], [1], [5], [0], [-14], [0], [0], [35], [-13], [37], [0], [0], [-35], [0], [7], [0], [-19], [0], [-35], [-28], [0], [0], [41], [0], [-23], [0], [0], [25], [0], [43], [0], [0], [0], [35], [29], [40], [0], [0], [0], [31], [-35], [0], [0], [0], [49], [-49], [-10], [0], [-49], [-44], [0], [-7], [0], [0], [0], [49], [-37], [0], [0], [-49], [0], [-22], [0], [-49], [7], [-53], [0], [-41], [26], [0], [49], [0], [56], [0], [0], [56], [0], [52], [0], [7], [0], [-35], [0], [56], [0], [-34], [0], [-47], [0], [-59], [0], [1], [0], [-35], [0], [0], [0], [61], [0], [0], [0], [-17], [49], [-19], [0], [0], [-14], [0], [53], [-7], [0], [-23], [0], [-28], [49], [55], [0], [65], [0], [0], [0], [-46], [35], [29], [0], [7], [0], [64], [0], [0], [0], [59], [50], [0], [0], [0], [-49], [70], [-14], [0], [0], [-28], [0], [0], [56], [71], [0], [-68], [0], [0], [49], [4], [0], [0], [0], [-43], [-7], [0], [-20], [0], [-58], [0], [0], [35], [0], [-67], [56], [-47], [0], [77], [0], [-77], [0], [0], [0], [0], [-7], [77], [62], [0], [0], [35], [0], [-32], [0], [-79], [0], [0], [-77], [0], [0], [53], [-49], [0], [70], [73], [0], [7], [-35], [0], [77], [0], [0], [40], [0], [49], [-80], [-77], [-83], [-14], [-28], [0], [59], [-1], [0], [0], [-11], [0], [49], [56], [0], [17], [0], [77], [0], [0], [0], [0], [0], [23], [0], [77], [0], [0], [-49], [70], [0], [52], [0], [-29], [89], [0], [-77], [0], [83], [0], [0], [0], [88], [0], [91], [0], [0], [-77], [0], [-91], [-85], [0], [37], [0], [91], [-71], [77], [0], [0], [41], [0], [-91], [0], [73], [0], [-82], [0], [7], [-91], [95], [0], [0], [-89], [0], [0], [91], [64], [0], [0], [56], [-10], [0], [-28], [-97], [0], [0], [0], [0], [0], [86], [0], [0], [0], [-91], [-22], [0], [-53], [26], [0], [-91], [0], [0], [-49], [-55], [0], [0], [-98], [0], [91], [0], [0], [100], [0], [56], [0], [-59], [70], [97], [0], [-103], [0], [0], [-49], [0], [38], [-28], [0], [61], [0], [-14], [0], [76], [0], [0], [-91], [-94], [0], [-7], [0], [-98], [0], [0], [-104], [65], [0], [0], [-91], [0], [0], [0], [-35], [0], [0], [70], [0], [0], [0], [56], [0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_7056_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_7056_2_a_bf();". // 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_7056_2_a_bf(:prec:=1) chi := MakeCharacter_7056_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_7056_2_a_bf();". // 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_7056_2_a_bf( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7056_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![0, 1]>,<11,R![0, 1]>,<13,R![-7, 1]>,<17,R![0, 1]>,<23,R![0, 1]>],Snew); return Vf; end function;