// Make newform 624.2.bv.e in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_624_bv();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_624_bv_Hecke();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_624_2_bv_e();". // 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_624_2_bv_e();". // 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 poly := [1, 0, -1, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]]; inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf); return Eltseq(inp_vec); end function; // To make the character of type GrpDrchElt, type "MakeCharacter_624_bv();" function MakeCharacter_624_bv() N := 624; order := 6; char_gens := [79, 469, 209, 145]; v := [6, 6, 6, 5]; // 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; // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_624_bv_Hecke();" function MakeCharacter_624_bv_Hecke(Kf) N := 624; order := 6; char_gens := [79, 469, 209, 145]; char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0], [1, 0, -1, 0]]; assert UnitGenerators(DirichletGroup(N)) eq char_gens; values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field F := Universe(values);// the Hecke field chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values); return chi; 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, 0, 1, 0], [1, 0, -2, -2], [-1, -1, -1, 0], [2, -3, -1, 3], [0, -3, 0, -1], [-4, -1, 4, 2], [-1, -3, -1, 0], [0, -3, 1, -3], [0, 2, 1, 2], [-2, 0, 4, -2], [-4, 7, 2, -7], [12, -1, -6, 1], [1, 5, -1, -10], [3, 0, -6, 3], [-3, -4, 0, 2], [0, 8, 0, 0], [4, 3, -4, -6], [14, 1, -7, -1], [-1, -3, -1, 0], [-1, 0, 2, -8], [6, -4, 0, 2], [3, 0, -6, 5], [-4, -6, 2, 6], [0, -6, 0, 0], [0, 4, -5, 4], [-17, 2, 0, -1], [0, 3, 5, 3], [2, 0, -4, 2], [10, -5, -10, 10], [0, 8, -4, 8], [-10, 4, 0, -2], [-4, -5, -4, 0], [4, 8, -4, -16], [3, 8, 3, 0], [1, 0, -2, -5], [-8, 18, 0, -9], [-2, -10, -2, 0], [4, 6, -2, -6], [-6, 6, 6, -12], [0, 11, -3, 11], [-14, -6, 0, 3], [0, -4, 0, 8], [-2, -10, 1, 10], [16, 4, -8, -4], [9, 3, -9, -6], [0, 6, 6, 6], [-8, -20, 4, 20], [3, 7, 3, 0], [8, 0, -16, 2], [-6, 16, 0, -8], [-5, 0, 10, -1], [-9, 2, -9, 0], [-10, -2, 10, 4], [0, -5, 18, -5], [0, -11, 9, -11], [-2, -2, 2, 4], [16, 8, -8, -8], [-4, -1, 4, 2], [-10, 0, 20, -5], [0, -5, -17, -5], [-13, -8, -13, 0], [-9, 0, 18, 7], [7, -10, 0, 5], [10, -4, 0, 2], [5, 0, -10, -12], [0, -20, 0, 0], [7, 16, 0, -8], [21, 7, -21, -14], [20, -2, -10, 2], [24, 1, -12, -1], [-7, 0, 14, -11], [0, 1, -13, 1], [12, 1, -12, -2], [-4, -2, 2, 2], [2, -2, 2, 0], [9, -24, 0, 12], [-6, -10, -6, 0], [-8, 15, 4, -15], [1, 16, 1, 0], [0, -2, -14, -2], [-16, 0, 32, -5], [14, 1, -7, -1], [-1, 8, 1, -16], [0, -5, 9, -5], [-26, 12, 0, -6], [-10, 6, -10, 0], [10, -10, -5, 10], [-1, 24, -1, 0], [11, 0, -22, 9], [3, 18, 0, -9], [-4, -30, 2, 30], [7, -9, 7, 0], [0, -1, -7, -1], [0, 0, 0, 32], [15, 15, -15, -30], [-2, -12, 1, 12], [12, 34, 0, -17], [0, -17, 7, -17], [10, 0, -20, 23], [-1, 6, 0, -3], [42, 6, -21, -6], [-28, -4, 28, 8], [0, 20, -4, 20], [-5, 22, 0, -11], [1, 0, -2, -2], [0, 16, 0, -16], [-8, 0, 16, -23], [6, -4, 0, 2], [0, -4, -1, -4], [-20, -12, 20, 24], [12, -1, -6, 1], [-14, 11, -14, 0], [2, 0, -4, -14], [-16, 20, -16, 0], [0, -15, 0, 30], [8, 0, 8, 0], [0, 14, -2, 14], [0, 2, 14, 2], [14, -14, -14, 28], [12, 1, -6, -1], [0, 4, 21, 4], [-42, -4, 0, 2], [8, -24, 8, 0], [10, 35, -5, -35], [-14, 20, 0, -10], [8, -27, 8, 0], [26, 2, -26, -4], [-5, -30, 0, 15], [0, 0, 0, -19], [-8, 4, 4, -4], [-12, -38, 6, 38], [0, 19, 17, 19], [0, -4, -14, -4], [-10, 6, -10, 0], [16, 30, -8, -30], [-4, -42, -4, 0], [-10, 22, -10, 0], [-34, 0, 34, 0], [0, 9, 38, 9], [-18, 0, 36, -14], [8, -6, -4, 6], [0, -24, 0, 48], [-2, 0, 4, -30], [0, -7, 0, -7], [8, 28, 8, 0], [18, 0, -36, -23], [-10, -34, 0, 17], [35, 14, 0, -7], [7, 0, -14, -5], [4, 17, 4, 0], [0, -23, 12, -23], [36, 8, 0, -4], [0, 8, 24, 8], [0, -6, -26, -6], [6, -4, 0, 2], [-6, 30, 6, -60], [-14, -15, -14, 0], [19, -28, 0, 14], [20, 0, -40, 22], [8, -28, -4, 28], [0, 24, 0, -48], [7, 0, -14, -31], [6, 14, -6, -28], [-8, -39, 4, 39], [12, 0, -24, 0], [0, 9, 7, 9], [-30, 5, 30, -10]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_624_bv_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_624_2_bv_e();". // 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_624_2_bv_e(:prec:=4) chi := MakeCharacter_624_bv(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); S := BaseChange(S, Kf); maxprec := NextPrime(997) - 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_624_2_bv_e();". // 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_624_2_bv_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_624_bv(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![1, 0, 14, 0, 1]>,<7,R![4, 12, 14, 6, 1]>],Snew); return Vf; end function;