// Make newform 912.2.bo.f in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_912_bo();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_912_bo_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_bo_f();". // 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_bo_f();". // 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, 0, -1, 0, 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_912_bo();" function MakeCharacter_912_bo() N := 912; order := 9; char_gens := [799, 229, 305, 97]; v := [9, 9, 9, 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_bo_Hecke();" function MakeCharacter_912_bo_Hecke(Kf) N := 912; order := 9; char_gens := [799, 229, 305, 97]; char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1]]; 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], [0, -1, 0, 0, 0, 0], [1, 0, -1, 1, 0, 2], [-1, 1, -1, 1, 0, 0], [0, -1, 0, 0, 0, -1], [-1, -1, 2, 1, 0, -2], [-2, 2, -3, 3, -2, 2], [2, 1, -2, 2, 1, 2], [2, -5, 0, 3, 3, 2], [2, -5, -2, -5, 2, 0], [-3, -3, 0, 3, 3, 3], [-3, -2, -2, 0, 6, -4], [-1, 3, -4, -3, 0, 3], [1, 0, -1, -3, -5, -2], [0, 3, -1, 3, 0, 0], [-2, -6, 0, 8, 8, -1], [-6, -4, -3, 3, 4, 6], [1, 0, 0, -1, -1, 11], [-4, 1, 4, 1, -4, 0], [-7, 0, 7, 1, 5, -6], [7, -1, 5, -2, 0, 2], [5, -5, 10, 5, 0, -5], [9, -2, -1, -9, 3, 3], [-7, -7, -5, 4, 3, 5], [-1, 4, 0, 0, -4, 1], [4, 4, 1, -8, 4, -1], [0, -1, -6, -1, -6, -1], [7, 3, -7, -7, 4, 4], [-2, 0, 2, 1, -13, -1], [-4, 5, 5, 0, 0, -5], [2, 2, 8, -6, 4, -8], [3, -2, -2, 2, 2, -3], [-8, 8, 0, 0, 0, -1], [3, 3, -5, -4, 1, 5], [3, 8, -2, -5, 0, 5], [-8, 14, 14, 0, -5, -9], [2, 0, -2, -5, -8, -3], [0, 5, -4, -16, -4, 5], [-5, 5, 0, 0, 0, -4], [5, 1, 2, -2, -1, -5], [0, 8, -8, -13, -8, 8], [13, -9, 4, -9, 13, 0], [0, -4, -4, 0, 9, -5], [-4, -5, 1, 5, 0, -5], [-1, -7, -4, 1, 11, 11], [2, 5, 10, 5, 2, 0], [0, 11, -3, 3, -11, 0], [-3, 0, 3, -1, 1, -4], [3, -3, -3, 0, 5, -2], [-14, 8, 8, 0, -4, -4], [2, 0, -2, -5, -8, -3], [0, -8, 14, -4, 14, -8], [10, 10, 5, -2, -8, -5], [-2, 3, 0, -1, -1, 13], [-15, 8, 4, 8, -15, 0], [10, 1, 3, -7, 0, 7], [11, 15, 6, -5, 0, 5], [3, 0, -3, 10, -3, 13], [0, 9, -3, 11, -3, 9], [3, -3, 0, 0, 0, 5], [5, 1, -6, 6, -1, -5], [0, -5, -8, 1, -8, -5], [5, 13, 2, -3, 0, 3], [-9, 8, -5, 9, -3, -3], [13, 7, -3, 7, 13, 0], [3, 3, 5, 2, -5, -5], [0, -20, 4, -9, 4, -20], [-5, 0, 5, 16, 2, 11], [2, 0, -2, 0, -5, 2], [-5, 0, 5, 5, -5, -5], [0, -6, 0, -27, 0, -6], [4, -8, 0, 0, 8, -4], [6, 6, -6, -5, -1, 6], [13, -10, -7, -13, 17, 17], [6, 0, 0, 0, -7, 7], [0, -9, -2, -2, 0, 2], [8, 3, 1, 3, 8, 0], [11, 0, 4, -4, 0, -11], [15, 5, 17, -17, -5, -15], [-14, 6, 10, 6, -14, 0], [7, 15, 15, 0, 4, -19], [-20, -10, -13, 7, 0, -7], [-13, -13, 1, 13, 0, -1], [20, -10, 0, -10, -10, 0], [-5, -23, -8, 8, 23, 5], [11, 11, 17, -8, -3, -17], [-12, 14, -4, 12, -10, -10], [-14, -19, -19, 0, 25, -6], [-8, 0, 8, 7, -15, -1], [3, -6, 9, -3, -3, -3], [0, -6, -3, 26, -3, -6], [-4, 11, 0, -7, -7, 18], [13, 2, -11, -13, 9, 9], [4, -11, -12, -16, 0, 16], [-9, 0, 9, 24, -5, 15], [-3, 15, -1, 15, -3, 0], [-6, -21, 0, 27, 27, 1], [0, 12, 1, 4, 1, 12], [-8, 21, -3, 21, -8, 0], [19, -1, 7, -1, 19, 0], [7, -14, 0, 7, 7, 17], [-2, -2, 15, -1, 3, -15], [-5, -2, 17, 5, -15, -15], [-3, 8, 8, 0, -16, 8], [-21, -11, -11, 0, 9, 2], [26, -11, 4, -26, 7, 7], [-22, -8, -3, 3, 8, 22], [14, -3, 0, -11, -11, -11], [-11, 3, 2, 3, -11, 0], [-5, 13, -19, 5, 6, 6], [-8, -17, -17, 0, 10, 7], [7, 0, -7, 1, -16, 8], [-8, 16, 0, 16, -8, 0], [0, 6, 10, 0, 10, 6], [-7, 13, 0, -6, -6, 3], [-21, 0, 21, 16, -6, -5], [2, -29, 4, 2, 0, -2], [7, -15, -15, 0, 13, 2], [-5, 21, -5, 5, -16, -16], [-6, -6, -13, 19, -13, 13], [5, -7, 0, 2, 2, -10], [0, 25, -21, -1, -21, 25], [-14, 8, -9, 14, 1, 1], [30, -4, -4, 0, 8, -4], [5, 12, -8, -5, -4, -4], [13, 13, 20, -20, -13, -13], [-6, -6, -6, -19, 25, 6], [-5, 2, -24, -19, 0, 19], [-7, 0, 7, -10, -11, -17], [0, 17, -16, 11, -16, 17], [-2, -22, 2, -2, 22, 2], [10, 21, 5, -5, -21, -10], [-2, 0, 6, 0, -2, 0], [6, 8, -14, -20, 0, 20], [3, -1, -1, 0, 27, -26], [-14, 15, 1, 15, -14, 0], [10, 10, -5, -22, 12, 5], [0, 21, -23, 3, -23, 21], [21, -5, -5, 0, -4, 9], [0, -12, -9, -12, -9, -12], [25, 25, 5, 2, -27, -5], [-3, -16, 0, 19, 19, 25], [-11, -11, -13, 26, -15, 13], [-20, -9, 10, -9, -20, 0], [13, 0, 0, -13, 0, 0], [-8, -13, -2, 6, 0, -6], [8, -22, -6, 6, 22, -8], [12, 8, -19, 19, -8, -12], [-10, 3, 0, 7, 7, 18], [0, 16, -27, -12, -27, 16], [1, 2, 18, 17, 0, -17], [4, 9, 13, -4, -22, -22], [18, 0, -3, 0, 18, 0], [11, 11, 2, 4, -15, -2], [-8, 0, 8, 6, -9, -2], [11, 4, 4, 0, -21, 17], [14, 1, 5, -14, -6, -6], [-26, 3, 5, -5, -3, 26], [24, 24, 7, -21, -3, -7], [-4, 3, -12, 3, -4, 0], [10, 13, 23, 13, 0, -13], [-8, 6, -28, -20, 0, 20], [-10, -17, 14, -14, 17, 10], [18, -14, 28, -28, 14, -18], [0, 5, 1, -23, 1, 5], [-4, 0, 4, 7, 34, 3], [22, -1, 12, -10, 0, 10], [-17, -11, 20, -11, -17, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_912_bo_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_912_2_bo_f();". // 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_bo_f(:prec:=6) chi := MakeCharacter_912_bo(); 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_bo_f();". // 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_bo_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_912_bo(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![81, -162, 162, -90, 36, -9, 1]>],Snew); return Vf; end function;