// Make newform 3234.2.a.bm in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3234_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_3234_2_a_bm();". // 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_3234_2_a_bm();". // 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 := [2, -4, -6, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, -4, -1, 1], [3, 2, -1, 0], [-3, 4, 2, -1]]; Rf_basisdens := [1, 1, 1, 1]; 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_3234_a();" function MakeCharacter_3234_a() N := 3234; order := 1; char_gens := [1079, 199, 2059]; v := [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_3234_a_Hecke(Kf) return MakeCharacter_3234_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 := [[1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 1], [0, 0, 0, 0], [1, 0, 0, 0], [2, 0, 1, -1], [1, 0, -2, -1], [3, 0, 1, 0], [-2, -1, 1, 1], [-2, -2, 0, 2], [1, -1, 0, -3], [2, 3, -1, -1], [3, 4, -2, 1], [-2, -3, -1, -1], [1, -2, -1, 0], [-2, 1, -3, -3], [0, -4, 2, 2], [6, 0, 3, 1], [-2, 3, 1, -1], [2, 0, 2, -4], [1, 4, 0, 1], [-2, -3, 3, 1], [1, 5, 0, 1], [6, -4, 1, -1], [0, -5, 0, -4], [6, -1, -2, 0], [1, -1, -4, -3], [-2, 5, -3, -3], [-2, 6, 2, -2], [-4, -4, -2, 2], [-2, -5, 1, -1], [11, -3, 2, 1], [0, 4, 2, -4], [-1, -2, -1, 2], [-8, -2, 2, -2], [-4, 6, -2, -2], [-9, 2, -2, -3], [0, 0, -4, -4], [-4, 4, 6, 6], [8, -5, 0, 4], [-2, 5, 3, 5], [-5, -12, -2, 1], [-14, 3, 1, 1], [-2, 12, 0, 4], [-8, -8, 2, 0], [-3, 5, 2, 7], [-2, 3, -3, 5], [-11, 1, 2, -1], [-11, 7, 2, -1], [1, 14, 0, 1], [-2, -8, -4, 4], [0, -10, -2, 2], [9, 4, 4, -3], [-2, 4, -6, -4], [6, 2, -1, 5], [-6, -1, -3, -1], [1, 4, 0, 1], [-8, -2, 0, 4], [-2, 4, 0, 6], [-10, -2, -2, 2], [-9, 2, -1, 2], [-12, -3, -6, -2], [13, -10, -3, -6], [-11, 0, -3, 2], [-2, 3, 8, 2], [-6, 3, 7, 7], [-2, -7, -5, -3], [14, -1, -1, 3], [2, -3, -3, -7], [0, -8, 3, 3], [2, 6, 1, -1], [6, 1, 3, -3], [-5, -1, -2, 5], [0, 0, -6, 0], [4, 12, 4, 4], [-21, 0, 3, -2], [-14, -6, 2, 2], [-5, 8, 0, 3], [4, -12, -2, 4], [17, -2, 0, -7], [-10, 8, -4, -6], [6, 0, 4, 8], [18, -1, 1, 3], [16, 3, -2, 6], [12, 8, -2, -2], [-10, -5, 9, 3], [-12, 0, 6, 2], [10, -2, -2, -6], [18, 5, -6, 4], [0, 8, 4, -4], [-16, -10, 2, -2], [-22, -4, 4, -6], [14, -5, -1, -1], [-4, -8, 0, -8], [2, 21, -1, 5], [-8, 8, 2, -10], [-7, 0, -4, 5], [2, -10, 1, 3], [-15, -10, -7, 6], [4, 0, 2, 0], [10, -6, 4, -6], [0, -8, -2, 0], [-1, -5, -4, -9], [-6, 16, -4, -4], [-6, -14, 0, -2], [6, 17, 0, 2], [14, 6, 0, 2], [11, -12, 2, 5], [18, -2, 4, 6], [19, 10, 4, -1], [-4, -6, 6, 2], [4, -2, 0, 0], [-2, 2, 6, -4], [-14, 2, 2, 4], [-6, -1, -1, 7], [-14, 6, -2, 0], [-4, 4, -2, -6], [-19, 8, -7, -10], [-22, 6, -2, -10], [-6, -15, -3, -3], [-1, -6, 0, -1], [-6, 7, -5, -1], [-12, -1, -2, -10], [-14, -3, 7, -7], [-14, -2, 8, 6], [-2, -16, 8, 0], [-2, -17, -5, 3], [-7, -2, -1, -4], [-3, 1, -2, 11], [10, -4, 11, 9], [-10, -10, 0, 2], [0, 10, 6, 2], [-14, 5, 1, -3], [22, 7, -5, -1], [27, 12, 8, -5], [15, -18, 0, -9], [11, 14, 4, -1], [-11, 12, 7, 0], [-7, 0, -10, -1], [-10, 7, -5, -1], [3, -20, -7, -4], [4, 22, -2, 6], [-2, -19, 1, 1], [8, 12, -4, 8], [13, 6, 0, -11], [-9, -6, -7, 4], [-18, 10, -3, -9], [1, 6, -2, 15], [0, 14, 4, -4], [2, -25, -7, -3], [10, 24, 4, 0], [0, 10, 9, -3], [-8, 8, 4, -8], [-28, 6, -2, 10], [22, -19, 3, 9], [-10, 11, -3, 13], [16, -10, -2, 6], [2, -4, -3, 19], [11, -22, -2, 1], [4, -17, 2, 2], [-28, 4, 0, 8], [-10, 5, 1, 17], [4, -14, 6, 10], [-2, 14, 6, 12], [-10, -14, -2, 0], [27, -2, 1, -4], [-4, 2, -6, 14], [16, -4, -5, 3], [-10, -12, -12, 0], [-5, -2, -2, -15], [-2, 1, -11, -15], [19, 20, -8, 3], [-8, 4, 0, 4], [2, 5, 9, 9], [-11, -7, 6, -9], [12, -2, -9, 7], [-2, 2, -12, -10], [-22, -8, -4, 0], [-18, 6, -4, -6], [4, -2, -1, 19], [8, 10, -10, 6], [17, -13, -10, -5], [38, 13, 1, -3], [19, -12, 8, 11], [-22, 4, 10, 4], [3, 16, 4, -9], [28, 20, 14, -4], [28, 8, -4, 4], [2, -3, -3, 1], [-10, -4, 6, -4], [-2, -5, -6, 4], [-14, 3, -1, -5], [22, -11, 3, 1], [35, 10, -4, -1], [4, -18, 6, 6], [22, -8, 3, 5], [-6, 7, 7, -9], [-2, -31, 1, 5], [29, -2, 4, -7], [-18, 14, -8, -10], [0, -2, -14, -2], [44, 8, 2, 2], [-23, 0, -4, -3], [-15, -2, 6, 7], [-28, -2, 6, 6], [-5, -22, 2, -3], [-9, -23, 8, -9], [12, -18, 10, 10], [26, -1, 3, -13], [16, -14, -2, 10], [34, 13, 1, 5], [-7, -22, 8, -7], [4, 36, -4, 0], [4, -2, 0, -4], [44, 10, 0, 0], [10, 9, -16, 2], [-2, -1, -11, 3], [25, 8, -8, -7], [6, -4, -10, 12], [2, 20, 0, -12], [-12, -24, 2, -2], [-12, 6, 4, -12], [12, -18, 0, 16], [20, -22, -6, -2], [21, 9, -4, -3], [6, -12, -16, -12], [-20, -14, -1, 3], [22, 2, -4, 2], [10, -18, 12, 14], [2, 17, 5, 1], [2, 27, -5, 7], [7, 2, -1, 10], [-36, 16, 0, 12], [26, 9, 1, 1], [-6, 14, 10, 12], [12, 12, -2, -2], [44, -3, -2, -6], [-4, 10, -10, 14], [-8, -18, -6, -6], [29, -8, 9, 2], [-2, 13, 3, 17], [25, -12, 3, -4], [-1, -11, 8, 7], [14, -6, -2, -10], [-5, -24, -8, -5], [-30, -16, 14, 0], [-33, 11, 6, 9], [-12, -12, 2, -2], [-18, 6, 12, -2], [26, 9, -9, 5], [-20, -8, 10, -8], [11, 14, -4, 7], [6, 2, -12, 10], [-24, -3, -10, 14], [12, 1, 0, -12], [46, 9, 1, 5], [30, 5, -7, 13], [-5, -4, -3, -8], [15, 4, 4, -5], [31, 2, 2, 1], [-20, 16, -8, -4], [0, 0, 16, -12], [-14, -14, -3, 11], [8, 0, 3, 19], [-4, 2, 9, 9], [-6, 0, -2, 12], [26, -27, 9, 13], [24, 6, -7, -7], [-2, 14, 20, 14], [-22, 18, -4, -10], [15, 10, -2, 13], [32, -16, 0, 0], [46, 7, 8, -2], [-46, 3, 1, -1], [-19, -14, 6, 3], [-7, -19, 14, 7], [28, 2, 2, -6], [28, 20, 14, -4], [-10, 10, 10, 6], [19, 11, 4, -13], [-26, -12, 0, 2], [0, 18, 2, 10], [10, -24, 4, 10], [12, 10, -19, -11], [-33, 6, 3, -6], [14, -17, 5, 5], [30, 16, -10, 4], [12, -8, -16, 4], [-15, -3, -14, 7], [1, 28, -4, 5], [-2, -21, 3, -13], [30, -22, -6, -8], [26, -2, -14, -2], [9, -31, 8, 5], [-14, -8, 4, 14], [-10, -5, 0, -18], [3, -37, 2, 9], [31, -28, 0, -9], [51, -1, 0, 7], [19, 20, -3, 0], [5, 6, 8, -11], [-38, -16, 4, 0], [-42, -7, 7, 1], [38, 5, 7, 1], [-16, 30, 2, 14], [20, -34, 0, 14], [-28, 0, -8, -4], [-41, 12, -6, -15], [10, 8, 18, 4], [18, -8, -8, -6], [-6, 30, 2, 0], [54, 10, -6, 6], [38, 6, -14, 2], [30, -3, -5, -3], [20, 12, 0, -8], [5, -4, 8, -11], [30, -16, 6, 0], [14, -7, 19, 5], [-2, -35, -8, -6], [10, -8, -16, -4], [11, 24, 9, 4], [-12, -16, 8, 12], [16, -1, -4, 4], [-10, 21, -3, -7], [22, -15, -3, -15], [-2, 5, 8, -2], [-8, -2, -2, 6], [28, 4, -16, 0], [-38, -7, -1, 13], [-2, 0, -4, -8], [-44, 12, -2, -12], [-18, 6, -2, -2], [8, 16, -6, -2], [3, 9, 10, -7], [-14, -8, 6, 12], [24, 18, -10, 10], [34, 5, 1, -3], [15, 6, -14, 17], [40, 13, 4, 0], [34, 0, -16, -10], [-34, -6, 6, -2], [2, 14, -6, 12], [16, -7, 4, 0], [-42, 9, 5, 13], [22, -14, -2, -18], [12, -12, 4, -28], [-13, -6, 8, 19], [8, 20, 0, 8], [8, 2, -2, -10], [-27, 4, -18, -13], [61, 16, 3, 0], [-6, -28, 2, 8], [34, 5, -11, -7], [-4, -34, 2, -10], [-30, 16, 0, -10], [19, 20, 4, 7], [-33, -8, 3, -6], [5, 19, 2, 11], [14, 21, 17, -11], [-14, 12, 22, 8], [38, 12, 12, -2], [-1, -16, -20, -5], [14, -16, -14, -16], [16, 26, -6, -2], [18, 16, 8, -6], [-8, -6, -2, -14], [-28, -19, 12, 0], [14, 13, 27, 5], [-14, -12, -16, -8], [4, 8, -16, -16], [-42, -28, -8, 6], [34, -29, 1, -9], [-16, 12, -2, 20], [8, -4, 12, 28], [6, 14, -10, 18], [-40, -4, -8, -12], [-30, 1, -5, 13], [-9, 14, 0, -17], [9, -28, 2, -13], [13, -14, 10, -17], [-29, 24, -14, -19], [-2, -6, -6, -6], [-22, 23, 5, -5], [-14, -1, 11, 23], [11, -30, -1, -6], [-17, -8, -9, 6], [22, -11, -11, -7], [-19, -28, 10, 7], [-10, 31, 1, -11], [23, 2, 13, -8], [-16, -10, 22, 2], [-59, 6, 4, -7], [32, -18, -2, -6], [30, 14, -16, 2], [2, -2, -16, 18], [10, -15, -3, -7], [-20, 36, -8, -12], [-12, 32, 3, -13], [-22, -25, -15, -9], [30, -8, -4, -16], [-2, 34, 1, 3], [22, -20, 4, -4], [0, -6, 18, 6], [-24, 8, 17, 5], [24, -27, 10, 2], [4, 16, -2, 6], [26, -2, 6, 6], [-26, -23, 17, 5], [-24, -14, 14, -14], [12, -16, -16, -24], [-11, 18, 6, 3], [-26, 10, 8, 2], [36, -2, 14, 14], [-42, 10, 12, 14], [34, 21, 17, -11], [31, -6, 12, 11], [18, -5, -3, -1], [-29, 1, 12, 3], [12, 8, -19, -7], [12, 16, 4, -16], [-38, -6, 6, -18], [6, 10, -11, 23], [10, 14, 8, 2], [-25, 17, 14, 13], [-34, -3, -22, -16], [-11, 30, 8, -3], [-18, -15, -21, 9], [14, -8, 8, -8], [-12, 14, -10, -2], [-22, -24, -4, -6]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3234_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3234_2_a_bm();". // 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_3234_2_a_bm(:prec:=4) chi := MakeCharacter_3234_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_3234_2_a_bm();". // 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_3234_2_a_bm( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3234_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![8, 16, -4, -4, 1]>,<13,R![-28, 80, -4, -8, 1]>,<17,R![776, 112, -52, -4, 1]>],Snew); return Vf; end function;