// Make newform 912.2.bn.l in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_912_bn();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_912_bn_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_912_2_bn_l();". // 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_912_2_bn_l();". // 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 := [9, -3, -2, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, -2, 2, 1], [3, 2, 0, -1]]; Rf_basisdens := [1, 1, 6, 2]; 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_912_bn();" function MakeCharacter_912_bn() N := 912; order := 6; char_gens := [799, 229, 305, 97]; v := [6, 6, 3, 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_912_bn_Hecke();" function MakeCharacter_912_bn_Hecke(Kf) N := 912; order := 6; char_gens := [799, 229, 305, 97]; char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, 0, 0], [0, 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, 1, 0, -1], [2, -1, 1, 0], [0, -1, -1, -1], [-2, 2, 2, -2], [2, 0, -1, 0], [-2, -1, -3, 0], [-3, 0, -2, 0], [3, 0, -2, -1], [-1, 2, 8, -1], [0, -3, 3, 3], [-2, 3, 1, -3], [0, 3, -3, -6], [-3, -2, 5, 4], [-5, 0, 2, -1], [-1, 2, -4, -1], [4, 1, -5, -2], [2, -4, 3, 2], [2, 0, -1, 0], [0, -3, 3, 6], [-9, 2, 7, -4], [-3, 6, 3, 0], [2, 2, -6, -2], [-1, 2, -4, -1], [-6, 9, 3, 0], [-9, 0, 2, -5], [-8, 3, 13, -3], [-4, -4, -4, -4], [6, -3, 3, 0], [2, 2, 2, 2], [3, 0, 0, 3], [2, 1, 3, 0], [-13, 0, 10, 7], [0, 0, 17, 0], [-6, 11, 5, 0], [2, 0, -4, 0], [7, -6, -1, 12], [4, -3, -3, -3], [-5, 10, 4, -5], [8, -1, -7, 2], [12, 0, 0, 0], [-7, 0, 2, -3], [10, -10, -10, 10], [-7, 0, -7, 0], [-8, 4, 12, -4], [-2, 4, 3, -2], [-1, 12, 11, 0], [-7, 6, -1, 0], [4, -8, -8, -8], [-6, 1, 1, 1], [6, -1, 5, 0], [-2, 2, 2, -2], [-4, 0, 5, 6], [-21, 0, 10, -1], [3, -6, -12, 3], [10, 1, 11, 0], [-4, -1, 5, 2], [-4, 3, 1, -6], [18, 4, 4, 4], [-5, 10, 4, -5], [-8, -1, 9, 2], [6, -6, -6, -6], [-8, -3, -11, 0], [-10, 2, 18, -2], [3, -6, -14, 3], [3, -6, -12, 3], [4, 3, -11, -3], [5, 0, 5, 0], [-2, 1, -1, 0], [10, 5, 5, 5], [4, -8, 0, 8], [-14, 1, -13, 0], [0, 0, -7, 0], [12, 0, -24, 0], [4, -9, 1, 9], [4, 1, -5, -2], [15, 0, -10, -5], [1, 0, -1, 0], [4, 7, -11, -14], [-31, 0, 14, -3], [-10, 2, 18, -2], [14, -15, -1, 0], [3, -6, 24, 3], [-10, 0, 11, 12], [1, -6, -5, 0], [27, 0, -14, -1], [-14, -2, -2, -2], [2, -9, -9, -9], [10, -13, -3, 0], [12, -1, -1, -1], [6, -10, -2, 10], [-21, 0, 10, -1], [18, -12, -24, 12], [-14, 1, -13, 0], [-3, -2, 5, 4], [15, 0, -14, -13], [-9, 18, 0, -9], [-26, -2, -2, -2], [18, 0, -9, 0], [-6, 12, -17, -6], [-8, 0, 1, -6], [19, 0, -6, 7], [-4, -4, -4, -4], [-38, -2, -2, -2], [4, -3, -3, -3], [6, -8, -8, -8], [-2, 13, 11, 0], [15, 0, -10, -5], [-1, 2, -28, -1], [2, -9, 5, 9], [4, -3, -5, 3], [30, -1, -29, 2], [31, 0, -18, -5], [28, 3, 3, 3], [-8, 16, 9, -8], [0, -9, 9, 18], [-9, 4, 5, -8], [10, 2, -22, -2], [8, -8, -8, 8], [-9, 18, -12, -9], [-3, 0, -6, -15], [2, 3, -7, -3], [30, -6, -6, -6], [0, 12, 12, 12], [4, -12, -12, -12], [2, -13, -11, 0], [-6, 12, -5, -6], [-2, 13, 11, 0], [23, 0, -23, 0], [2, 0, 0, 0], [-13, 0, 13, 0], [-36, -3, 39, 6], [8, 0, 5, 18], [-23, 0, 23, 0], [4, -8, 0, 8], [10, -20, -1, 10], [-5, 10, 16, -5], [0, 9, -9, -9], [-18, -6, -6, -6], [-20, 16, 24, -16], [11, 0, -16, -21], [7, 0, -6, -5], [-3, 6, -4, -3], [7, -14, 16, 7], [10, -9, -11, 9], [36, -3, -33, 6], [13, 12, -25, -24], [-8, 7, 1, -14], [-14, 28, 3, -14], [12, -12, -12, -12], [9, 12, 21, 0], [12, -8, -16, 8], [2, -4, 31, 2], [3, -6, 36, 3], [-20, 9, -11, 0], [0, 0, 0, 0], [-8, -3, -3, -3], [-22, 23, 1, 0], [-12, 24, 13, -12], [-5, 10, 40, -5], [-14, 13, -1, 0], [28, -5, -23, 10], [-1, 12, -11, -24], [32, 5, -37, -10], [34, -2, -2, -2], [-4, 17, -13, -34], [-29, 12, -17, 0], [6, -12, -17, 6]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_912_bn_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_912_2_bn_l();". // 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_912_2_bn_l(:prec:=4) chi := MakeCharacter_912_bn(); 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_912_2_bn_l();". // 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_912_2_bn_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_912_bn(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![16, -36, 31, -9, 1]>,<7,R![-8, 1, 1]>,<17,R![256, 240, 91, 15, 1]>],Snew); return Vf; end function;