// Make newform 3174.2.a.bb in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3174_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_3174_2_a_bb();". // 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_3174_2_a_bb();". // 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, 3, -4, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [-2, 0, 1, 0, 0], [0, -3, 0, 1, 0], [2, 0, -4, 0, 1]]; Rf_basisdens := [1, 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_3174_a();" function MakeCharacter_3174_a() N := 3174; order := 1; char_gens := [2117, 1063]; v := [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_3174_a_Hecke(Kf) return MakeCharacter_3174_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, 0], [-1, 0, 0, 0, 0], [1, -2, 1, 0, 1], [-2, 1, 1, 0, 1], [-4, 2, -2, 2, -3], [1, 2, -3, 1, -1], [1, -1, 0, -3, 0], [-2, -1, 1, -2, 1], [0, 0, 0, 0, 0], [2, -1, 2, -3, 3], [-4, 0, -2, -1, -1], [-3, 1, 0, 3, -1], [-4, 0, -2, 3, 1], [-4, 0, -3, 2, 3], [0, 1, 4, -2, -1], [-3, 6, -4, 2, -4], [-3, 5, -2, 7, -3], [-2, 2, -1, 1, -7], [-3, -5, 4, -4, 1], [-1, 1, 3, 1, 5], [-1, -5, 3, 4, -1], [-8, -2, -3, 0, 1], [-1, 0, 2, -1, 0], [-1, 6, -2, -4, 1], [-4, 7, 1, 0, 3], [5, -2, 8, -6, 11], [-10, 1, -5, 7, -8], [-1, -8, 6, -6, 4], [-3, -2, 0, -6, -2], [-6, 0, -9, -1, 3], [-5, -4, 5, 1, -3], [3, 4, 6, -1, -5], [3, -1, 10, 0, -1], [10, -2, 7, 0, -2], [-5, -1, -2, 11, -3], [10, -11, 5, -12, 8], [-1, -4, -1, -1, 2], [2, -9, -3, -3, 10], [-1, -3, -3, 3, -2], [-5, 1, -7, -5, -4], [-7, 0, -4, -9, -1], [3, 1, 3, 5, 2], [3, -2, 2, -13, 8], [3, 3, 3, 10, -4], [3, -1, 6, -4, 4], [-10, 11, 5, 6, -2], [4, -5, 12, -5, 14], [8, -1, -2, -7, 2], [-1, 1, -10, 5, 4], [-4, 3, -7, 4, -3], [-2, 8, -5, 9, -10], [-7, 5, -10, 12, -8], [-7, 3, -14, 7, 0], [8, 3, 0, 2, -3], [-5, 0, -7, 11, -9], [10, -8, 8, -7, 1], [-10, 3, -1, 8, -6], [2, -3, -5, 5, -2], [-12, -4, -4, 3, -3], [-7, 10, -8, 11, -7], [0, -2, 7, -9, -8], [13, -4, 3, 0, 7], [-8, 13, -4, 9, -5], [-1, -4, -7, 5, -1], [14, -16, 13, -11, 18], [-5, 6, -6, 5, -12], [4, 4, 6, -3, 2], [-2, 10, -2, 5, 0], [3, 1, -7, 0, 6], [-1, 8, 6, -8, 3], [16, -13, 3, -7, 6], [-10, 3, -16, 14, -6], [4, -9, 7, 5, -1], [16, -10, 7, -9, 12], [3, -9, 1, 1, 3], [19, 0, 5, -4, 1], [14, -9, 10, -6, 1], [-8, 20, 5, 2, -8], [4, -5, 4, 3, -1], [9, -4, 9, -9, 23], [13, -10, -2, -11, 2], [-1, -2, 4, 13, 6], [-6, -3, 1, 0, 0], [-2, 18, -6, 8, 0], [-13, 2, -4, -2, 1], [-7, -10, -6, -1, 6], [2, -2, 19, 4, 0], [6, -9, 10, 5, -4], [14, -14, 20, -15, 14], [-7, 13, 0, -1, -1], [9, -2, -11, 5, 4], [2, -5, 3, 0, 2], [-20, 1, 3, 3, 7], [8, 2, 0, 10, -1], [14, -14, 12, -6, 17], [-5, 3, 6, 3, 2], [9, -18, 7, -19, 16], [-10, 1, -11, -9, 6], [-14, 12, -12, 15, -20], [3, 6, 5, 2, -20], [5, 12, -7, 1, 1], [4, 17, -13, 11, -17], [9, 2, 4, 8, 19], [-4, 8, -20, -3, -11], [-9, -3, 8, -10, -8], [3, -22, 17, -10, 8], [24, -21, 10, -13, 16], [-12, 0, -4, 3, 8], [8, -26, 1, -11, 12], [15, -16, -2, -6, -1], [-3, -12, -5, 14, -9], [10, 3, 18, -5, -8], [26, -8, 10, -12, 14], [-2, 8, 4, -9, 5], [-18, -1, -12, 9, -5], [7, -14, 12, -19, 25], [-16, 6, -16, 7, 7], [-9, 2, -10, 16, -5], [-12, 12, -18, 2, -11], [14, 12, 3, 2, -1], [-13, 11, -18, 11, -2], [-3, 5, 4, -17, 0], [14, 12, -7, 9, -2], [1, 8, 12, -3, -4], [-5, 0, -16, 9, -1], [-1, 16, -6, 7, 4], [-1, 12, 12, -9, 4], [-25, 2, -2, -7, -16], [5, 1, -6, -6, -7], [13, -3, -5, -6, -2], [9, -11, 18, -12, 27], [10, -23, 10, -21, 8], [-26, -8, -3, 1, 1], [26, -2, -2, 2, -14], [-19, 13, -8, 31, -11], [1, 20, -12, 16, -15], [6, 4, 3, 9, 4], [-2, -12, 12, -15, 0], [31, -2, -9, 0, -4], [-20, 28, -6, 5, -14], [-26, 6, -19, 19, 0], [8, 8, 1, -17, 5], [18, -9, -12, 3, -19], [5, 2, -8, 4, 5], [12, -3, 4, -19, 12], [3, 5, 18, -6, 13], [-17, 15, -14, 2, -12], [21, -5, 6, -6, -5], [-3, -8, 19, 17, 4], [-26, 30, -21, 28, -24], [2, 4, 1, 3, 16], [-24, 11, -16, 13, -25], [-13, 15, -8, 16, -2], [-4, 3, -5, -14, -17], [-1, -18, 32, -8, 13], [-9, 1, -12, 4, 7], [10, -11, 11, -19, 12], [-12, 3, 15, -1, -1], [19, -20, 18, -14, 10], [-30, 16, -19, 17, -6], [14, -1, -4, -21, -3], [-20, 4, -29, 7, -5], [-15, 10, -16, 1, -20], [-24, 1, -4, -3, -17], [10, 9, -7, -7, 0], [19, 5, 10, -3, -14], [-5, 15, -5, -3, -3], [-20, 7, 6, 12, -3], [-2, -28, 18, -20, 12], [1, 7, 7, -12, 0], [3, 5, 1, 5, -13], [-4, -9, 7, -30, -3], [-8, -25, 8, -2, 12], [-20, -2, 4, 0, -18], [1, 0, 4, -19, 4], [-7, 0, -1, 15, -17], [-11, -13, 5, -7, -1], [1, 7, -4, 18, -20], [-19, 14, -3, 27, -2], [31, 0, 20, -17, 14], [-16, 28, -21, 13, -24], [-10, 11, 0, 12, -16], [10, -15, 10, -16, 0], [0, -17, 4, -14, -16], [-4, 2, 4, -9, 4], [27, 1, 15, -13, 31], [10, 17, -7, -2, 10], [-12, -10, -17, -6, 13], [-13, 19, -16, 29, -13], [-6, 4, -19, 9, 4], [-1, -7, -3, -33, 1], [6, -23, 0, -5, -6], [-26, -4, 1, 14, -5], [-14, -8, -12, -6, -3], [-8, -2, -2, -7, -1], [-8, 14, 2, 27, -1], [-37, 5, -5, 10, -7], [-5, -13, -3, -10, 7], [38, -14, 34, -16, 17], [0, 12, -6, 3, 9], [-31, 7, -2, 25, -9], [5, -5, -15, 6, 11], [-24, 17, 11, 3, 4], [0, -3, -10, 10, -20], [12, -11, 19, -9, 13], [-3, 25, -38, 9, -18], [28, -8, 12, -17, 13], [6, -24, 14, -21, -5], [-42, 18, -23, 14, -21], [-32, 21, -30, 25, -13], [15, 2, 11, -11, 23], [0, 6, -7, 1, 4], [-23, 25, -12, 8, -19], [12, 9, -19, -10, -12], [-12, -12, 19, 3, 5], [-2, 4, 22, -7, 35], [1, -15, -2, -9, -23], [14, -10, 35, 0, 10], [-13, 15, 6, 10, -12], [27, -34, 25, -8, 29], [8, 13, 0, -3, 25], [17, -21, -8, -1, -1], [5, -3, 3, -12, -16], [-3, 11, -43, 11, -4], [3, -3, -6, 8, 21], [0, 3, -5, 15, -19], [19, 10, -6, 7, -6], [-9, -5, 1, -1, 15], [5, 9, -2, -6, -10], [-4, 31, 9, 1, 0], [-4, -29, 19, -10, 4], [18, 20, -17, -1, 7], [-27, -10, -7, 9, -7], [16, -19, 38, -21, 30], [9, 12, -17, 19, -34], [6, 1, -4, 27, 10], [-12, 22, -30, 27, -11], [-1, -6, -12, 2, -40], [-18, 21, -5, 9, 1], [8, -18, 9, -8, -6], [-11, 21, -16, 6, -2], [-2, -5, 1, 16, 2], [25, 22, -20, -2, 3], [27, -11, 0, -4, -2], [-28, -3, 13, -9, 15], [18, 7, 0, 17, 17], [11, -10, 32, -7, 3], [-38, -10, -5, 15, -2], [2, -28, 29, -18, 10], [-31, 27, -22, 23, -7], [16, 2, 10, 25, -22], [-31, -6, 8, -15, -16], [24, -35, 15, -22, 27], [30, 4, 17, -6, -3], [15, 12, 2, 17, -29], [-45, 19, -15, -8, -33], [-12, 14, -12, 15, -26], [12, 3, 3, -17, 19], [-10, 3, 8, 29, 4], [-16, -10, 18, -1, 0], [-20, 32, -25, 14, -12], [29, -6, 5, -11, -7], [-44, 17, -14, -6, -34], [26, -15, -7, 10, 6], [-35, 12, -15, 19, 12], [24, -10, 2, -6, 11], [-42, 8, 2, -10, 13], [9, -25, -9, -5, 17], [21, -34, 29, -30, 25], [1, -4, -36, 6, -22], [-16, -2, -11, -7, -16], [6, 18, -8, 6, 16], [18, -1, 18, -6, 14], [19, 4, 5, -19, 11], [-16, -6, -19, 20, -4], [-4, 24, -11, -23, 4], [-36, 20, -26, 20, -8], [-28, 20, -26, 31, -24], [-28, 10, -19, 26, -35], [-1, -11, 1, 8, 3], [8, -2, 11, -12, 28], [-60, 7, -2, 2, -8], [-7, 20, 1, 4, -28], [17, -6, -10, 10, 4], [-14, 4, -19, -14, -21], [-10, 13, -4, 20, -5], [12, -29, 15, -44, 35], [-10, 10, -17, -11, 8], [-14, 20, -6, -27, -7], [7, -3, -16, 14, -36], [-19, 12, -4, 18, -15], [-23, -9, -15, 1, -14], [-31, 12, -4, 27, -22], [-2, -31, 29, -29, 20], [43, -3, 21, -20, 15], [6, -31, 13, -18, 5], [-31, 4, -2, -2, -6], [2, 19, -47, 14, -22], [27, -50, 8, -28, 22], [-6, -38, 23, -11, 12], [20, -18, 7, -21, 20], [-14, -42, 8, 4, 3], [-5, 13, -17, -5, -21], [-15, -9, -13, -1, -1], [4, 0, 35, -13, 10], [-2, 19, -7, 11, 22], [51, -3, 25, -29, 24], [-26, 15, -2, 5, -33], [16, -23, 40, -20, 37], [21, -33, 14, -8, 9], [11, 17, 20, -16, 30], [-46, 15, -34, 5, -9], [-50, -4, 3, -15, -1], [33, -6, -4, 10, 10], [-36, -5, -12, 20, -1], [32, -13, 8, 9, 9], [-27, 30, -4, -4, -25], [-3, 9, -24, -8, 21], [-11, 3, -33, 9, -14], [1, -5, 3, -40, 8], [-25, 0, -6, -21, -12], [-18, -7, 20, -9, 29], [2, 9, -36, 1, -9], [-10, 6, 12, 5, -3], [-9, 22, -1, 12, 32], [18, -3, -11, 9, 3], [-1, 42, -32, 38, -15], [-1, -20, 2, -21, -21], [0, 15, -24, 12, -37], [16, -13, 33, -30, 24], [3, -8, -13, 18, 12], [-3, -2, -14, 4, -1], [20, 8, -4, -5, -14], [-25, 30, -5, 25, -20], [-10, 7, -25, 16, -6], [30, -8, 13, 9, -13], [26, 17, -8, 2, -4], [10, -22, 40, -17, 38], [6, 7, -13, -15, -20], [-5, 20, 8, 24, 14], [-7, -6, 9, 35, -6], [5, -7, -20, -2, -25], [37, -46, 10, -3, 23], [4, -4, -4, -3, 35], [42, -39, 33, -30, 42], [20, 13, -26, 18, 9], [-3, 4, 6, 13, -19], [-11, 12, 2, -22, -19], [1, -15, -31, 6, 6], [20, -10, 16, 24, 2], [2, 38, -32, 24, -30], [1, 0, 18, -5, 31], [-3, -8, -13, 19, -6], [-9, -20, 32, 14, 3], [30, -4, 20, -48, 25], [47, -31, 30, -5, 21], [27, -20, 32, 8, 6], [-53, 43, -9, 20, -31], [-9, 37, -3, 19, -18], [-38, -3, -35, 36, -8], [13, -17, -13, -19, -13], [-8, 9, -15, 6, 6], [-3, 22, -43, 23, 0], [-30, 35, 9, 5, 9], [-5, -5, 6, -20, 9], [6, 6, 20, -12, 6], [17, 26, 13, -31, 0], [3, 16, -1, -15, 5], [13, 13, -19, 18, -36], [-17, -16, 21, -22, 8], [7, 29, -18, 6, -18], [-38, 50, -2, -2, -29], [-13, 21, -2, -32, 8], [22, -27, 20, -31, -10], [-13, -32, 0, -13, 29], [10, 25, -15, -1, -12], [-31, 20, -5, 55, -12], [-43, 39, -7, 0, -33], [-41, 14, -2, -1, -24], [11, 42, -12, 6, -20], [8, -13, -14, -32, -6], [-27, -21, 5, 2, -28], [-44, -18, 0, -11, 3], [6, -24, 42, -24, 47], [-13, 26, -43, 16, -12], [8, -28, 20, -1, 3], [8, 11, 2, -29, -10], [-44, 1, 18, 8, 9], [9, 20, -9, 10, 12], [45, -20, 41, -8, 12], [-23, 30, -32, 52, -8], [-8, -1, 5, 1, -2], [53, -14, 20, -17, 21], [6, -4, 3, 4, 8], [14, -40, 8, -30, 5], [50, -5, 34, -45, 44], [-14, -13, 2, -26, 4], [25, -20, 15, -17, 47], [-5, -28, 22, -18, 24], [0, 4, 22, -4, -8], [12, -22, 21, -15, 9], [-34, 27, 12, 1, 23], [-42, 7, -30, 32, -3], [24, -3, 2, -8, -9], [56, 18, -5, 0, -1], [-1, 25, 11, 16, -14], [7, 14, -17, -18, 18], [18, -13, 20, -12, 16], [-13, 16, -16, 4, -46], [-2, -22, 23, -23, -7], [9, -20, 23, 2, -22], [-48, 19, -12, 30, -16], [-22, 0, -30, 29, 17], [-41, 56, -32, 54, -43], [-32, 3, -6, -32, 37], [37, -30, -14, -38, 1], [12, 23, -11, 19, -34], [-58, 33, 9, 11, -1], [18, 5, 22, -29, 2], [38, -42, -9, -16, 18], [-29, -17, -5, -1, -23], [23, 36, -20, 31, -13], [-41, 33, 3, 2, -35], [45, -8, 22, -26, 10], [-7, 18, -16, 26, -12], [35, -35, -5, 1, 40], [9, 5, 17, 22, 31], [38, -24, -27, -14, -33], [8, 22, -1, -13, 7], [39, -21, 2, 1, -18]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3174_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3174_2_a_bb();". // 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_3174_2_a_bb(:prec:=5) chi := MakeCharacter_3174_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_3174_2_a_bb();". // 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_3174_2_a_bb( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3174_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-1, 3, 14, -15, -1, 1]>,<7,R![-253, -231, -22, 33, 11, 1]>],Snew); return Vf; end function;