// Make newform 3174.2.a.ba in Magma, downloaded from the LMFDB on 19 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_ba();". // 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_ba();". // 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], [0, 0, 1, 1, 1], [3, 0, 1, -2, 1], [2, -1, 0, 0, -2], [4, -2, 2, -1, 3], [-1, 0, -3, 1, 0], [3, 0, -1, 0, 1], [0, 0, 0, 0, 0], [0, -1, 1, 1, -2], [-2, -1, 3, -2, 2], [3, -1, 3, -1, 0], [-2, -3, -1, -2, 2], [1, 6, -1, 3, -3], [-4, 5, -2, 5, -4], [-1, 0, -2, -2, -4], [-1, 1, -5, 3, 2], [1, -6, 0, -1, -1], [7, -3, 0, 1, 4], [-4, -2, -4, 4, -3], [-4, 4, -7, -2, -3], [5, 4, -3, 5, -3], [3, -2, 1, -2, 2], [-1, 3, -6, -4, -2], [5, 2, 1, -8, 1], [-3, -3, -2, 6, -8], [5, -3, 2, 4, -5], [7, -2, 0, 2, 6], [3, -2, -6, 2, 0], [-3, 12, -10, 9, -9], [-10, 8, -6, 1, -5], [-3, 11, -5, 10, -6], [7, -11, 10, -9, 10], [3, 9, -7, 5, -7], [3, -1, 9, 3, -2], [5, -3, 7, -6, -5], [0, 3, -2, 5, -1], [5, -13, 6, -12, 3], [2, -1, 0, -6, 3], [2, -3, 12, -6, 7], [-3, -3, 13, -4, 4], [0, -1, 8, -4, 3], [-1, 6, -11, 0, 2], [0, 7, -13, 6, -3], [-3, 2, -2, 5, -6], [15, -7, 11, -16, 5], [-8, -2, -7, 7, -12], [10, -4, 9, -3, 2], [-9, 14, -5, 9, -10], [-3, 4, -3, 4, -7], [3, 5, -4, 3, 5], [3, -2, -2, -5, 10], [-7, 14, -7, 11, -14], [-8, -3, 2, -3, 0], [2, 2, -4, -7, 7], [-2, -7, 1, 0, 8], [-9, 5, -7, 2, 1], [7, -3, 0, -8, 5], [-8, -1, 1, -8, 4], [-1, 1, 3, -2, -8], [7, -15, -2, -5, 7], [-10, 4, 3, 1, 3], [-4, 1, -5, 9, 4], [6, -6, 2, -11, 7], [-1, 5, 2, 3, 13], [1, 6, 1, 0, 6], [-2, 4, -3, 10, -6], [0, 2, 3, -8, -2], [10, -13, 7, -6, 7], [-7, 3, 2, 14, -6], [13, -3, 4, -10, -3], [-6, 10, -2, 13, -16], [3, -8, 12, 2, 7], [-9, 5, -2, 3, 7], [-2, 2, 2, 8, 1], [-14, -4, 1, -5, 5], [-4, -9, 4, -1, 10], [-13, 13, -7, 25, -5], [0, -5, 7, 1, 4], [0, -14, 0, 5, -9], [-15, 4, -13, 12, -2], [5, 2, 17, -2, 4], [7, -1, 1, 2, 1], [-4, 6, 2, -12, -6], [-9, -5, 6, -2, 4], [-1, -12, 7, -16, 6], [-17, 19, -23, 17, -19], [4, -14, 15, -1, 10], [-6, 6, -5, 6, -20], [-7, 1, 1, 13, 0], [-20, 15, -6, 13, -11], [1, -1, 3, 2, 3], [-23, -4, -6, 4, -3], [8, 1, -10, 2, 0], [2, -5, -6, -2, -12], [11, -4, 9, -9, 6], [2, -9, 12, -11, -7], [-1, 17, -20, 10, -11], [2, -8, 3, 0, -12], [-2, 25, -7, 11, -5], [12, -8, 6, 5, 7], [-17, -4, -2, -4, -13], [-5, 15, 12, -6, 4], [-16, 9, -23, 12, -20], [17, -16, -2, -5, 8], [-14, 9, -7, -5, -17], [14, -6, 3, -11, -10], [-8, -12, 1, -4, 4], [7, -11, 10, -25, -1], [17, -1, 8, -18, 2], [2, 4, -9, -17, 5], [8, -26, 13, -21, 18], [-16, 4, -2, -2, 10], [6, 1, -5, -12, 4], [6, 7, -3, 13, -12], [5, 13, -7, 2, 12], [0, 23, -9, 10, -16], [1, -5, -6, -8, 10], [6, -7, 16, -6, 18], [-11, -4, 5, -15, 3], [-5, 16, -7, 7, -18], [-7, 4, 13, 9, -4], [-21, 5, 2, -5, -7], [-11, 16, -9, 20, -12], [11, -15, 7, -16, 16], [-5, 10, 1, -10, -6], [13, -8, 3, -24, 12], [-23, 14, 9, 0, 2], [-11, -1, -12, 5, -6], [-18, 3, -11, 8, -5], [-9, -9, -6, 7, -18], [0, -2, -11, 13, 10], [23, 4, -2, 11, -3], [-28, -12, 0, 4, -2], [-11, 3, -23, 5, 8], [-13, -3, 4, -8, -12], [-3, 1, 12, -7, 3], [14, -12, -3, 0, 12], [-40, 5, -9, 11, -9], [-14, 8, 1, 22, 6], [-7, -19, 0, -13, 19], [7, -4, 16, 9, -1], [30, 7, 9, -21, 12], [-13, 13, -4, 6, -8], [8, -8, 15, 1, -4], [15, -5, 12, -23, 18], [-3, -2, 12, 1, 14], [15, 11, 0, 1, -6], [-22, 15, -36, 11, -19], [-5, 3, -7, 9, 21], [1, -15, -4, 5, -1], [8, -9, -3, 5, -16], [-5, -6, -8, 7, 8], [1, 12, 19, -2, 5], [33, -19, 24, -14, 32], [-3, 19, -8, 11, -12], [1, 1, -8, 0, 11], [-27, 16, -14, 18, -15], [-1, -8, 4, 2, 18], [11, 13, -2, 3, -19], [18, -1, 25, -5, 4], [-9, 24, -22, 25, -29], [1, 4, 15, -6, 16], [20, -13, -7, 3, -4], [-17, 7, -14, -2, -7], [-9, -24, 7, -15, 10], [0, -2, 8, 10, 5], [-26, 9, -18, 13, -6], [20, -6, -2, 10, 18], [-6, 7, 5, 14, -7], [-2, -14, 6, -6, 1], [-11, 10, 23, -2, -7], [16, 4, 6, 17, 8], [24, -22, 4, -2, 4], [-3, 0, 15, 4, -4], [6, -16, 14, 1, -1], [-16, 6, 2, -8, -5], [5, 16, -14, 3, 4], [16, 1, 24, -11, -3], [-11, -6, 3, -20, 20], [5, 3, 8, 7, 21], [10, -16, 12, -11, 0], [0, 10, 6, -5, -10], [-4, 20, 10, -13, -4], [8, 0, -5, -6, 4], [-12, 16, 2, -16, 15], [17, -17, 9, 10, 7], [-5, 30, -23, 27, -17], [3, -3, -13, 3, 16], [13, -23, 10, -15, 19], [2, -4, 36, -10, 3], [6, 6, 5, -23, 0], [27, -6, 15, 3, 1], [-2, -9, 18, -20, 12], [6, 1, -9, 4, -2], [10, -3, 29, -16, 2], [32, -2, 5, 0, -5], [2, 10, -13, 16, -3], [-4, -17, 18, -20, 34], [6, -15, 3, 6, 6], [29, -7, 23, -5, -2], [20, -26, 9, -20, 15], [-35, 7, -14, 28, -11], [-10, -10, 0, 13, -10], [7, -6, 10, -8, 19], [35, -20, 29, -13, 38], [-16, 1, -5, -4, 12], [-8, 19, 7, -10, -14], [-19, -2, 9, -5, 23], [-2, -17, 5, -9, 30], [4, -12, 0, 13, -11], [7, -11, 6, -1, 7], [11, -7, -4, -13, -12], [-31, 7, -29, 10, -19], [-31, 14, -22, 7, -19], [24, 13, 15, -26, 22], [3, 21, 11, -17, 2], [-21, 25, -35, 25, -35], [19, -18, 16, -21, 6], [2, -4, -17, -9, -25], [8, -25, 3, 13, 0], [-25, 7, -9, 29, -8], [2, 19, 9, 0, -3], [-40, 39, -32, 32, -43], [9, -27, -2, -9, 6], [5, 14, -10, -2, 5], [-25, 0, 1, -4, -6], [-10, -14, 0, -4, -1], [-7, -8, -8, -7, -2], [-13, 9, -10, 40, -9], [-23, 15, -9, -10, -19], [-35, 24, -18, -3, -17], [20, 0, 2, 17, -7], [-22, 8, -17, 19, -38], [-26, -17, 2, 5, -17], [-10, 14, 23, 3, -4], [-18, 19, -3, 8, -30], [-11, -28, -10, 18, -12], [-13, -6, -4, 16, 5], [-1, 15, -1, -9, -9], [-5, 14, -10, -5, -16], [-3, -1, -17, -4, -1], [45, -23, 22, 2, 20], [27, 2, 4, -11, 0], [-41, -2, -4, 10, -13], [18, -17, -17, 7, 0], [-21, 29, -25, 22, -32], [33, 3, 10, 15, -5], [27, -19, 11, -1, 29], [9, 15, 1, -5, -22], [-6, -32, 35, -12, 10], [39, -24, -7, -2, 8], [-9, 12, -7, 20, 15], [-13, -20, 11, -21, 17], [13, 31, -19, 14, -2], [-30, 18, 23, 4, 15], [0, -14, 3, -2, -12], [-9, 16, -14, -6, 3], [-18, 4, -37, 11, -8], [-34, 18, -17, 8, -18], [-5, 13, -11, -7, -25], [-24, -12, -6, 1, 5], [-30, 20, 20, 3, 14], [-33, 13, 3, 22, -7], [-20, -27, -4, -3, 15], [-22, 9, -4, 8, 2], [44, 11, -8, -10, 2], [-18, 26, -14, 34, -9], [8, -4, -1, 5, 29], [37, -14, 30, -40, 36], [-5, 5, 18, -13, 11], [-14, 24, -2, -10, -8], [0, -4, 12, -17, 18], [-14, 6, -14, -9, 5], [3, -15, -1, -25, 19], [7, -15, 34, 13, 11], [-10, -18, 6, -6, 26], [-2, -2, -5, -6, 26], [9, -16, 7, 9, -19], [2, 2, 9, 10, 1], [-3, -17, 1, 9, -11], [58, -6, 0, -5, -2], [8, -29, 5, -21, 1], [-27, 14, 0, 16, -10], [5, 2, 33, -15, 19], [-6, 1, -16, 9, 4], [3, 20, -29, 14, 15], [-7, 25, -28, 7, -17], [-8, 1, 33, 14, 6], [23, 20, 2, -19, 16], [15, -11, 14, -8, -4], [8, 1, -14, 24, -15], [-27, 18, -23, 8, 4], [31, -9, 0, 2, 29], [22, 6, -1, 18, -21], [7, -8, -5, 18, 13], [29, -4, -4, -2, -2], [49, -25, 33, -28, 47], [19, -14, 20, -42, -8], [-29, 11, -12, -15, -23], [-13, 13, -14, 11, 7], [22, -5, 12, 34, 8], [-12, -4, -22, 4, -17], [-2, -12, 14, -22, 13], [31, -25, 22, -35, 35], [5, -29, -4, 12, 7], [26, 1, 4, 22, -25], [24, -31, 3, -13, -2], [24, -3, 20, -17, 40], [7, 5, -6, -19, -14], [-9, -10, -4, 37, -20], [12, 25, -29, 19, -34], [53, -4, -12, 1, 3], [37, -14, -6, -10, 4], [24, 11, 8, 17, -12], [-24, 1, 17, 5, 8], [-23, 21, 8, 26, 4], [21, -45, 32, -15, 24], [-22, 19, -24, 30, -33], [-2, -5, 37, -2, -3], [19, -6, -27, 6, -6], [38, 9, 11, -13, 20], [38, -27, 35, -27, 36], [-22, 15, -17, 18, -12], [8, 33, 11, -21, -1], [-29, 14, -2, 14, -11], [-31, 17, 6, -10, -32], [-3, 23, 19, -18, -2], [-24, -13, -12, 9, -24], [17, -9, 3, -20, 33], [16, -25, -5, -21, 13], [11, -13, 10, -16, 14], [24, 10, 9, 4, 4], [-20, 15, -20, 25, 5], [-15, 19, -9, 18, -25], [17, 26, -22, 5, -13], [-34, 4, -6, -9, -8], [30, -2, 23, -18, 40], [19, 7, 28, -6, 13], [13, 6, 32, -28, 8], [-16, 15, -44, 3, -9], [-25, -5, -22, 27, -20], [27, -13, -7, -36, -10], [-8, 39, -7, 8, -4], [-9, 9, 3, 6, 33], [46, -35, 8, -13, 26], [-9, 25, -19, 10, -6], [-13, 21, 20, 14, -2], [-32, 37, -25, 46, -31], [-4, -14, 40, -6, 16], [34, -2, 8, 6, 32], [-17, -13, -13, 18, -18], [10, -7, -6, -21, 13], [41, -29, 46, -12, 32], [-10, 5, -28, -16, 20], [17, 9, -25, -1, -30], [5, -26, 40, -12, 32], [44, -22, 11, -34, -9], [-6, 15, -16, 34, 3], [-3, -27, -1, -38, 35], [-26, 0, -32, 30, -13], [7, -21, 9, -6, 15], [40, -43, 20, -21, 43], [39, 0, 14, -44, 9], [-11, -3, 14, 1, -6], [-14, 14, -8, 26, -20], [4, 13, 18, 39, -13], [-4, 6, -16, -15, -1], [-32, -17, -1, 6, -19], [-38, 13, 1, 5, -21], [25, 0, 12, 11, 18], [-36, 27, 4, 48, 2], [11, 10, -34, -19, -2], [-2, -30, -11, 7, 20], [-13, -29, 13, -32, 0], [25, -3, 16, 10, 15], [26, -7, 50, -15, -5], [36, -26, -7, -32, -7], [39, -22, -3, -12, -2], [23, 8, 6, 30, 12], [-22, 8, -46, 27, -14], [32, -33, 7, 16, 5], [-44, -3, 11, -18, 0], [-36, -5, -18, 18, -42], [-30, 31, -27, 17, -43], [-12, 17, -19, -8, -20], [6, 12, 27, 13, -2], [-62, 9, -26, 19, -18], [-18, 21, 1, -11, -9], [-4, -29, 33, -21, 41], [-9, 24, 20, 2, -32], [-13, 7, -6, 4, -5], [33, -1, -3, 6, -20], [3, -5, -7, -1, -3], [-6, -3, -22, 32, 8], [-16, 10, -11, -29, 34], [16, 2, -24, 11, 2], [-10, 32, -2, 5, 15], [27, 2, 4, 6, 22], [-22, 30, -18, 26, -22], [-9, 12, -6, -1, -21], [-46, -11, -13, 39, -12], [12, 27, 2, 23, -30], [-22, -11, -6, 1, 2], [61, -4, 5, 13, 5], [-12, 25, -27, 36, -11], [-24, 35, -35, 3, -17], [-2, 4, -8, 7, -20], [-3, -30, -12, 0, -16], [-25, 30, 0, 1, -23], [-14, 45, -25, 3, -23], [-36, 4, -18, 7, 12], [-8, 47, -1, 30, -30], [9, -11, 22, -24, -32], [26, 43, -38, 3, -6], [51, -15, 52, -44, 14], [23, 23, -8, 12, 11], [-67, 10, -20, 42, -9], [4, -20, -7, -27, 22], [-47, 27, -25, 51, -9], [24, -18, -6, 22, -5], [-43, 7, 11, -16, -20], [-44, 38, -5, 36, -3], [23, 12, 4, 14, -22], [9, -4, -10, 2, 16], [40, -45, 4, -40, 5], [8, 14, 39, -22, 17], [65, 6, 41, -51, 27], [9, -8, 14, 21, 1], [37, 20, -3, -19, -2]]; 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_ba();". // 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_ba(: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_ba();". // 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_ba( : 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;