// Make newform 637.2.g.k in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_637_g();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_637_g_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_637_2_g_k();". // 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_637_2_g_k();". // 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, 3, 15, -16, 38, 0, 7, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [1249, -8, -36, -602, -44, -76, -7, -12], [193, 38, 171, 2287, 209, 361, -24, 57], [-375, 2724, -5375, 7125, -361, 1375, -250, 193], [-363, 5383, -5203, 6897, -507, 1331, -242, 174], [-2933, -250, -1125, -13889, -1375, -2375, 182, -375], [534, -3617, 7654, -10146, 602, -1958, 356, -273]]; Rf_basisdens := [1, 1, 458, 916, 916, 458, 916, 458]; Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf); Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]]; 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_637_g();" function MakeCharacter_637_g() N := 637; order := 3; char_gens := [248, 197]; v := [2, 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; // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_637_g_Hecke();" function MakeCharacter_637_g_Hecke(Kf) N := 637; order := 3; char_gens := [248, 197]; char_values := [[-1, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 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, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0], [2, 1, 0, 1, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0], [0, 1, 1, 0, -1, -1, 0, -1], [1, -1, -1, -1, 1, 1, -1, 1], [0, 0, -1, -3, 0, 0, -2, 0], [0, 1, 0, 0, 0, 1, 0, 1], [0, 0, -1, 0, 0, 1, -1, 1], [0, 0, 0, 0, -1, 0, 0, -1], [0, 1, 0, 0, -2, 1, 0, 0], [6, 2, 1, 2, 6, 0, 0, -1], [0, 1, 0, 0, 1, 0, 0, 0], [-1, -3, 0, -3, -1, 1, -1, 0], [0, -2, 0, 0, 1, 4, 0, 2], [1, -2, -1, -2, 1, -2, 2, 1], [2, 0, -1, -2, 0, 0, -2, 0], [-1, 0, 1, 6, 0, 0, 4, 0], [0, 0, 0, 0, -4, -2, 0, -2], [0, -2, 0, 0, -2, 2, 0, 3], [6, 1, -1, 1, 6, -3, 3, 1], [-1, 0, -1, -4, 0, 0, 0, 0], [0, -1, 0, 0, 0, 2, 0, -1], [0, -2, 0, 0, -1, -5, 0, -1], [-2, 0, 0, 3, 0, 0, 2, 0], [-5, -1, -2, -1, -5, 3, -3, 2], [0, 1, 0, 0, 6, 0, 0, -1], [0, 3, 0, 0, 7, -3, 0, 0], [0, 4, 0, 0, -5, 0, 0, 2], [-5, 2, 1, 2, -5, -5, 5, -1], [9, 0, 1, 0, 9, -3, 3, -1], [10, -2, -3, -2, 10, 5, -5, 3], [0, -5, 0, 0, 3, 2, 0, -2], [-4, 0, -3, -2, 0, 0, 1, 0], [0, 1, 0, 0, -3, -5, 0, -2], [0, 8, 0, 0, 3, -7, 0, -2], [10, 0, -1, 5, 0, 0, -3, 0], [7, 4, -1, 4, 7, 4, -4, 1], [-7, 0, 0, -1, 0, 0, 3, 0], [1, 0, 4, -3, 0, 0, -1, 0], [2, 0, -1, 4, 0, 0, -2, 0], [-6, 0, -2, -6, 0, 0, -3, 0], [-8, 0, -4, -5, 0, 0, -3, 0], [16, -5, -1, -5, 16, 4, -4, 1], [3, 3, 2, 3, 3, 1, -1, -2], [7, 6, 1, 6, 7, 4, -4, -1], [-5, 2, -3, 2, -5, -3, 3, 3], [9, -4, -1, -4, 9, -2, 2, 1], [-2, 0, 0, 0, -2, 2, -2, 0], [4, 11, 3, 11, 4, -8, 8, -3], [-1, 0, -1, -2, 0, 0, 0, 0], [-1, -4, -2, -4, -1, 1, -1, 2], [0, -4, 0, 0, -2, 6, 0, -4], [0, -6, 0, 0, 3, -1, 0, 2], [13, 0, 5, 6, 0, 0, 2, 0], [-9, 6, 3, 6, -9, -1, 1, -3], [0, 6, 0, 0, 7, -8, 0, -1], [1, -10, -4, -10, 1, 2, -2, 4], [-8, 0, 0, -1, 0, 0, 7, 0], [4, 0, 1, 11, 0, 0, 5, 0], [0, -6, 0, 0, -4, 0, 0, 3], [9, 0, -2, -1, 0, 0, 2, 0], [0, -5, 0, 0, -9, -4, 0, 4], [-6, 0, 2, 0, -6, -2, 2, -2], [7, -6, -4, -6, 7, 4, -4, 4], [0, 0, 4, 0, 0, 0, 3, 0], [-2, 0, 7, 6, 0, 0, 1, 0], [2, -3, -3, -3, 2, 7, -7, 3], [-11, -11, -4, -11, -11, -3, 3, 4], [-16, 0, 4, 5, 0, 0, 6, 0], [-5, 4, 5, 4, -5, -1, 1, -5], [4, 0, 1, -15, 0, 0, -5, 0], [15, 0, 5, 7, 0, 0, 0, 0], [-12, 4, 2, 4, -12, 3, -3, -2], [-1, 0, -4, 5, 0, 0, 2, 0], [0, 3, 0, 0, -4, -5, 0, 2], [-2, 0, -4, -4, 0, 0, 4, 0], [0, -2, 0, 0, -24, 1, 0, 3], [-8, -9, 0, -9, -8, -4, 4, 0], [-9, -2, 3, -2, -9, -4, 4, -3], [-12, 0, 2, 1, 0, 0, -5, 0], [-3, 0, -7, -2, 0, 0, -3, 0], [0, 1, 0, 0, -13, 3, 0, 3], [0, 5, 0, 0, -7, -2, 0, 2], [4, -4, -4, -4, 4, -6, 6, 4], [0, -14, 0, 0, -6, 6, 0, 4], [0, -1, 0, 0, 22, 3, 0, 0], [0, -9, 0, 0, 10, 8, 0, 2], [11, 0, 3, 2, 0, 0, 4, 0], [5, 12, 5, 12, 5, 4, -4, -5], [6, 0, -8, -6, 0, 0, -3, 0], [-1, 6, 3, 6, -1, 3, -3, -3], [3, 14, 7, 14, 3, -3, 3, -7], [-3, -7, 2, -7, -3, -4, 4, -2], [0, 10, 0, 0, -4, 3, 0, 4], [0, -13, 0, 0, -3, 15, 0, 3], [0, -6, 0, 0, -6, 0, 0, 5], [0, 2, 0, 0, -10, -4, 0, 0], [-18, 2, -1, 2, -18, -3, 3, 1], [-7, 0, -1, -2, 0, 0, 2, 0], [11, 0, -1, 5, 0, 0, 6, 0], [-20, -3, -5, -3, -20, 1, -1, 5], [0, 10, 0, 0, -6, -10, 0, 0], [0, 5, 0, 0, -2, -6, 0, -7], [0, 3, 0, 0, -19, 5, 0, 3], [5, 10, 9, 10, 5, -5, 5, -9], [-7, -8, 6, -8, -7, -5, 5, -6], [0, 6, 0, 0, 3, -11, 0, -5], [16, 7, -4, 7, 16, 2, -2, 4], [-9, 0, 0, -7, 0, 0, -7, 0], [-13, 0, 7, -1, 0, 0, -6, 0], [0, -14, 0, 0, -7, 6, 0, 6], [0, -7, 0, 0, -20, 1, 0, 1], [0, -3, 0, 0, 24, 0, 0, -1], [7, 11, -3, 11, 7, -6, 6, 3], [0, -3, 0, 0, -5, -1, 0, 2], [6, 0, -4, 4, 0, 0, -10, 0], [0, -12, 0, 0, -1, -9, 0, -1], [0, 4, 0, 0, -6, -8, 0, 2], [6, 0, -5, 2, 0, 0, 4, 0], [-7, 12, 6, 12, -7, -8, 8, -6], [-4, 5, -5, 5, -4, 6, -6, 5], [23, 19, 6, 19, 23, -5, 5, -6], [0, 1, 0, 0, -4, 6, 0, -5], [18, 0, -2, -4, 0, 0, -2, 0], [-6, 0, 3, -4, 0, 0, 9, 0], [13, 0, 3, -10, 0, 0, -2, 0], [-9, 0, -2, 9, 0, 0, -1, 0], [5, 2, 2, 2, 5, -1, 1, -2], [2, 0, -2, 16, 0, 0, 2, 0], [-11, 0, -1, 0, -11, 11, -11, 1], [0, 0, 0, 0, -5, 6, 0, -9], [16, 8, 0, 8, 16, -8, 8, 0], [7, 0, -1, 12, 0, 0, 9, 0], [-11, -13, -4, -13, -11, -3, 3, 4], [21, -3, 6, -3, 21, -1, 1, -6], [0, -5, 7, -5, 0, 3, -3, -7], [0, -4, 0, 0, 6, 6, 0, -4], [-21, 0, 8, 6, 0, 0, 6, 0], [16, 0, 2, 2, 0, 0, -1, 0], [0, 9, 0, 0, -5, 1, 0, 4], [-7, 3, 6, 3, -7, 7, -7, -6], [-15, 0, -4, 5, 0, 0, 4, 0], [-10, 0, 4, 11, 0, 0, -4, 0], [0, 9, 0, 0, -14, -8, 0, 3], [17, 0, -7, -7, 0, 0, -13, 0], [0, -14, 0, 0, -13, -1, 0, 0], [8, -1, 7, -1, 8, -5, 5, -7], [0, 9, -7, 9, 0, -3, 3, 7], [-11, 0, 2, -12, 0, 0, -2, 0], [17, -9, -5, -9, 17, 3, -3, 5], [-3, 0, -9, 2, 0, 0, 6, 0], [0, -18, 0, 0, -14, 8, 0, 4], [-14, 0, 1, -1, 0, 0, 9, 0], [24, 0, 1, 5, 0, 0, 12, 0], [6, 0, 0, -4, 0, 0, -12, 0], [5, 0, 1, 17, 0, 0, -5, 0], [11, 0, 5, -9, 0, 0, -3, 0], [0, 17, 0, 0, -9, 1, 0, 4], [0, -6, 0, 0, -14, -1, 0, -2], [0, 7, 0, 0, -3, -9, 0, 10], [29, 0, -3, -4, 0, 0, -4, 0], [-23, 0, 6, 1, 0, 0, -7, 0], [-33, 0, -4, 2, 0, 0, 2, 0], [0, 12, 0, 0, -3, 1, 0, -5], [14, 0, -8, -8, 0, 0, -9, 0], [-7, 11, 3, 11, -7, -3, 3, -3]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_637_g_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_637_2_g_k();". // 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_637_2_g_k(:prec:=8) chi := MakeCharacter_637_g(); 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_637_2_g_k();". // 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_637_2_g_k( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_637_g(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![1, 3, 15, -16, 38, 0, 7, -1, 1]>,<3,R![-6, 16, -9, -1, 1]>,<5,R![81, -9, 109, -114, 160, -86, 37, -7, 1]>],Snew); return Vf; end function;