// Make newform 2415.2.a.w in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2415_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_2415_2_a_w();". // 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_2415_2_a_w();". // 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 := [-12, -89, 160, 244, -196, -143, 87, 30, -16, -2, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 1, 0, 0, 0, 0, 0, 0, 0], [6, 0, -7, 0, 1, 0, 0, 0, 0, 0], [-2, 37, -5, -123, -25, 72, 11, -15, -1, 1], [0, 47, -5, -125, -25, 72, 11, -15, -1, 1], [7, 57, -62, -158, 24, 84, -2, -16, 0, 1], [-14, -39, 91, 149, -35, -83, 3, 16, 0, -1], [16, 119, -131, -421, 23, 238, 7, -47, -1, 3], [-11, -103, 121, 335, -47, -178, 4, 33, 0, -2]]; Rf_basisdens := [1, 1, 1, 1, 2, 2, 1, 1, 2, 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_2415_a();" function MakeCharacter_2415_a() N := 2415; order := 1; char_gens := [806, 967, 346, 1891]; v := [1, 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_2415_a_Hecke(Kf) return MakeCharacter_2415_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 := [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 1, 0], [1, 0, 0, 0, -1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, -1, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 1, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -1, 0, 0, 0, 1, 0, -1, -1, -1], [1, -1, 0, 0, 0, 1, 0, 0, 0, 1], [2, -1, 0, 0, 1, -1, 0, -1, -1, 1], [-2, 1, -2, 0, 0, 1, 0, 0, -1, -1], [0, 0, 0, -1, -1, 1, 1, 0, 0, 0], [2, -1, 0, -1, 1, -2, -1, 0, 1, 1], [0, 1, 0, 1, 0, 0, 0, 0, -1, -1], [-3, 0, 0, 0, -1, 2, 1, 1, 0, 0], [1, -1, 0, 1, 0, 0, 1, 1, 0, -1], [-1, 0, -2, 0, 0, 0, 1, 0, 0, 0], [1, -1, 0, 0, 0, -1, 0, -1, 1, 1], [0, 0, 0, -1, -2, 1, 0, 0, 0, 0], [5, -1, 2, 0, 1, -1, -1, 1, 1, 1], [-3, 3, -2, 1, -1, 0, 1, 1, 1, -1], [2, -2, 2, 0, -1, 0, 1, -1, 1, 0], [3, -1, -2, 0, 2, -1, -2, -1, -1, 1], [-4, 0, -2, 1, 0, -1, 1, 0, -1, 0], [2, -3, 0, -1, 0, -2, 0, -1, 0, 1], [-1, 1, 0, 0, 1, 3, 1, 0, -2, -1], [1, 1, -2, 0, 1, 3, 1, -2, -2, -1], [-2, 2, -2, 0, 1, -2, -1, 0, 0, 2], [-1, 0, 0, 1, 2, -3, 0, 0, 1, 0], [-4, -2, -2, 1, 0, 1, 1, 1, -2, -2], [3, -2, 2, -1, -1, -1, -1, 0, 3, 2], [1, 1, 0, 0, 0, 1, 2, 2, 2, -1], [2, 0, 0, -1, 0, -3, 1, -2, 1, 0], [2, -1, -2, 0, 2, -3, 0, 0, 1, 1], [2, -3, 0, 1, 0, 0, 0, -1, -2, -1], [-1, 0, 0, 1, 2, 1, 0, -2, -1, 0], [-1, -4, 0, 1, 0, 1, 0, 0, -1, 0], [-4, 2, 0, 0, 0, 0, 0, 1, -1, 2], [3, -1, 2, 0, 2, -3, 0, -1, -1, 1], [5, 1, 0, 1, 0, 0, 0, 2, -2, -1], [-2, 0, 0, -1, 1, 5, 0, 0, -3, 0], [-1, 2, -2, 1, 0, 1, 0, 2, -1, -2], [-1, -3, 2, 0, 0, 1, 0, -1, 1, -1], [0, 1, 2, 1, 0, -4, 0, -1, 0, -1], [8, 1, 2, 0, 2, -1, 0, 0, 1, -1], [-4, 3, -6, 0, 1, 3, 0, -2, -2, 1], [-3, 0, 2, 0, 2, -2, -1, 2, 0, 0], [7, -6, 4, 0, 2, -2, 0, -1, -2, 0], [-4, 4, 0, 1, -2, 3, 0, 1, -1, 0], [3, -7, 0, 0, 2, -1, 0, 1, 1, -1], [2, -2, -4, -1, 2, 1, -1, 0, -3, 2], [2, -2, 0, 0, -2, 0, -2, -2, 2, 2], [4, -3, 2, 1, 1, -2, 1, -1, 0, -1], [3, 4, 4, -1, 0, -1, 0, 0, -1, 0], [-5, -1, 2, -1, -4, 4, 0, 2, 0, 1], [4, -8, 0, 0, 1, -2, -1, 0, 0, 0], [-5, 0, -4, 1, -4, 3, 0, 0, 1, 0], [2, -2, 0, -1, 3, 1, 1, 0, -2, -2], [5, 3, 4, 0, -2, -3, 0, -1, -1, -1], [-5, 1, -2, -1, -4, 0, 0, 0, 0, -1], [4, -4, 0, -1, 1, -1, 1, -2, 0, 0], [-12, 1, 0, -1, 1, 0, -1, 1, 4, -1], [2, -1, 0, -1, 2, 2, 0, 1, 0, 1], [2, -1, 4, -2, 0, -3, -1, 1, -1, 1], [8, -4, 0, 1, 4, -5, 1, -2, 1, 0], [-7, 2, 2, 0, -2, -2, 1, 1, 3, -2], [-5, 0, 0, -2, 2, 0, -1, 0, -2, 0], [9, 3, 2, -2, 3, -3, -1, 0, 2, 1], [3, 3, 6, 0, -1, -1, -1, 1, -1, 1], [4, 3, 2, 0, 0, -1, -1, 2, 0, -3], [-5, -2, 0, 1, -2, 5, 0, 0, -1, -2], [4, 4, 6, 0, -2, -2, 0, -1, 3, 0], [1, -1, -2, 0, -1, 5, 1, -1, -1, -3], [3, -2, 4, -2, -3, 2, 0, -1, 1, 0], [3, 4, 0, 2, 0, 2, 1, 0, 0, -4], [-6, -4, -4, -1, 0, 5, 2, 2, 0, 0], [10, -1, 2, -1, 3, 0, -1, 0, -1, 1], [3, 1, 2, -2, 3, 1, -1, 0, -2, 3], [2, -4, 2, 0, 2, -4, 0, 0, 2, 4], [2, 4, -2, 0, -6, 2, 2, 2, 0, 0], [-5, 3, -2, 3, 0, 0, 1, 0, -1, 1], [8, -1, 0, 1, 4, -6, -2, -1, 0, 1], [-11, 3, -2, 0, -2, -1, -2, 0, 2, 1], [-3, -3, 4, 2, 0, -1, 2, 1, 3, 1], [-3, -3, -2, 2, 0, 3, -2, 3, 1, 1], [-9, 3, 0, 0, -5, 5, 1, -1, 1, 1], [0, 5, 0, 0, -1, -3, 2, 0, 2, -1], [7, 0, 0, -1, 3, 1, -1, -5, -2, 2], [-1, -1, 4, -3, -2, 0, -2, 0, 2, 1], [-6, -6, 0, 2, 2, -6, -2, -2, -2, -2], [-4, 6, 0, 2, -3, 0, 1, 3, 1, 0], [5, -5, 2, 0, 0, 3, 0, -1, -1, -3], [-5, 8, 0, 0, -4, 4, 3, 0, 0, 0], [0, 10, -2, 0, -1, 2, 1, 1, -1, 0], [-6, -4, 2, 1, 2, -5, 1, 0, 3, 0], [-6, 4, 4, 2, -3, 2, 1, -1, 3, -2], [-1, 0, 2, -1, 0, 1, 0, 2, -1, -4], [8, -1, 2, 0, 3, -3, -1, 1, 4, 3], [2, -2, 4, 0, 3, 0, -3, -1, 1, 0], [-1, 5, 4, -2, -2, 1, 0, 0, -4, -1], [0, 0, 4, 0, 0, 0, 0, 2, -2, 0], [-2, 5, -6, 2, 0, 1, -2, 0, 1, -1], [-3, 3, -6, -2, 1, 7, 1, 1, -1, -3], [-10, 0, 4, -1, 2, -3, 4, 2, 0, 0], [-5, -2, 2, 1, -4, 3, 2, 0, -3, -2], [-7, 3, 2, 0, 3, 3, -3, 1, -1, 1], [-7, 1, 0, -2, -4, 1, 2, -1, -3, -1], [1, 3, -4, -2, 1, 5, -1, -4, -2, 1], [-1, -2, 0, 0, 2, -2, -3, -4, 2, 2], [-10, 2, 0, -3, -1, 3, -1, -1, 1, 2], [-1, -13, 2, 0, 0, 1, 0, 2, 0, 1], [-2, 7, 0, 2, -4, 1, 1, 1, 3, 1], [-4, 8, -4, 0, -4, 2, 0, 1, 3, 0], [14, 6, 4, -1, 2, -5, -3, -2, 5, 2], [6, 3, 2, -1, 2, -4, 0, 1, 4, 3], [-8, 0, 2, -2, 2, 0, -2, -2, -4, 0], [-6, -3, -8, 0, 2, 5, 1, -1, -1, -1], [0, -6, -2, 0, -3, -4, -1, -2, -2, 2], [2, 10, 0, 1, -4, 7, -1, 2, -3, -2], [5, 6, 0, -1, -2, -3, -4, 2, 1, -2], [-23, 6, -4, 2, -6, 8, 1, 2, 0, -2], [-3, 5, 0, 1, 0, 0, -2, 0, 0, -1], [2, -2, -6, 4, 3, -2, -1, 0, 0, -2], [7, -4, 0, -1, 0, -7, 0, -2, 3, 4], [2, 2, -6, -2, 2, 0, -2, -2, 2, 2], [-13, -5, 6, 1, 4, 0, -2, 1, 1, 1], [-6, 3, -4, 1, 2, 0, 0, -1, -8, -1], [1, -1, 6, -2, -2, -1, 0, 1, 9, 1], [9, 5, 2, 1, 2, 2, 0, 4, -2, 3], [10, -5, -6, 3, 7, -4, 1, 0, -1, 1], [-2, -2, 2, 0, -4, 2, 4, -1, -1, -2], [-14, 10, 0, 3, -3, 5, 1, 3, 1, -2], [-12, -3, -2, -1, 3, 2, -1, 0, 3, -1], [13, 8, 4, 0, 0, 2, -1, 4, 4, 0], [-3, 6, 0, -2, 0, -4, -1, 4, 2, 2], [8, -1, 4, -1, 2, 0, -2, -1, 0, 3], [1, -9, 6, -1, 2, -2, 2, 0, -4, 1], [4, 6, -2, 2, -4, 0, 2, -2, 2, -2], [6, 6, 6, -2, -3, -4, 1, 0, 4, 2], [0, 2, 2, 0, -6, 2, 2, 0, 2, 2], [-9, 0, 6, 1, 1, -3, 3, 1, -4, -2], [-16, -1, -4, 1, 1, 0, 1, 4, 1, -3], [11, 3, -2, 2, -3, 1, 1, -2, -2, -3], [6, 4, -4, 0, 2, 2, 0, 0, -4, 0], [1, 1, 4, 0, -6, -1, 0, 0, 6, -1], [2, -11, 4, -3, -1, 0, 1, -1, 2, 3], [-11, 3, 0, 2, 2, -1, 0, 0, -6, -3], [-7, 4, -8, 0, -4, -2, 1, 0, 0, 0], [-7, 4, 0, -3, 0, -3, -2, 2, 3, 0], [10, 8, 4, 0, -4, 6, 2, 2, -4, 0], [6, -4, -2, -2, 3, -2, -1, -5, -1, 2], [-3, -7, 8, 2, 3, -5, 1, -2, 0, 3], [-3, 9, -2, -1, -2, -2, 4, -2, 0, -1], [-9, 5, -2, 1, 1, 0, 1, 2, -6, -1], [12, -4, 4, -2, 2, 2, -2, -3, -1, 0], [10, -7, 4, -4, -1, 1, 0, 2, 0, 3], [16, 3, 6, -3, 4, 0, -2, -1, 2, 1], [0, 6, 0, 0, 1, 0, 1, 3, 3, 0], [10, -6, 0, 1, 2, -3, -2, -1, 5, 0], [-3, 4, -6, 1, -7, 5, 1, 0, -1, 0], [2, 0, 0, -2, 2, 0, -4, -4, 0, 4], [-12, 2, -2, 0, 0, -2, -2, 0, 2, 2], [2, -5, -6, 1, -2, 4, -2, 2, 3, 5], [-12, 2, -6, 0, 5, 6, 3, -3, -5, 0], [-6, -2, 4, -3, 0, -3, -3, 2, 1, 2], [8, 0, 6, 0, -3, -2, -1, 0, 8, 0], [1, 7, -10, 4, 3, -3, 3, 0, -6, -3], [4, -5, -4, 1, 7, -6, 1, 1, 4, 1], [1, -2, -2, -5, 1, 1, -1, -2, -5, 2], [-3, 11, -4, 2, -7, 1, 3, 0, -2, 1], [4, -7, -2, -2, 1, 3, 1, 0, 1, -5], [-3, 10, -6, -2, 0, 0, -3, -3, -3, 2], [18, -2, 2, -2, 3, -2, -3, 0, -2, 2], [14, 8, 2, 0, 2, -6, 2, -4, -2, -4], [-8, 6, -4, -1, -10, 7, -1, 2, 3, -2], [6, 16, 2, 0, -2, 0, -2, 6, 0, -4], [-11, 4, -2, 3, -2, 1, 4, 6, 3, -4], [-6, 7, -4, 4, -2, -3, 3, 1, 1, -1], [3, -4, -4, 1, 5, -1, -1, -1, -4, -2], [4, 8, -4, -4, -2, -2, -4, -6, 0, 0], [-9, 2, -2, 3, 2, 3, 2, 2, -9, -2], [-17, 8, -6, 3, 0, -1, 2, -3, -4, -4], [12, -11, 6, -3, -1, -2, -3, -2, 5, 3], [18, -10, -2, 2, 1, 2, 1, -1, -1, 4], [-5, 2, -4, 1, -6, 7, 2, -4, -1, -2], [-4, 4, 2, 2, -2, 0, 2, 6, 8, 0], [-8, 0, 2, 3, -2, 7, 3, 0, -5, -4], [-12, 2, 2, 0, 4, 0, 2, -3, -3, -2], [-8, 1, -2, 0, 2, -3, 1, 4, -2, 3], [0, 5, 0, 0, 0, -1, -1, 3, 3, 1], [8, -6, 2, 2, 2, 4, 2, -5, 1, 2], [-14, 13, -2, 1, 1, -8, -3, 2, 7, 3], [-7, 6, -2, 3, -6, 11, 2, -2, -5, -6], [19, -5, 2, -1, -4, -10, -4, 0, 6, 1], [5, -9, 0, 0, -3, 9, 1, 0, -8, -3], [-1, -10, -6, -2, 6, -4, -3, -5, 3, 2], [7, -5, -6, -3, 2, -4, -2, 0, 2, 5], [-15, 8, -4, -3, -6, 5, 0, 2, -3, 0], [7, 5, -6, -1, 1, 2, 2, 2, 3, -1], [-17, 9, 2, 2, -4, 7, -2, 0, 0, -5], [-1, -1, 0, 0, -4, -3, -2, -2, 0, -3], [12, 17, -4, 0, -2, 5, 2, 2, 1, -1], [-12, 10, -2, 2, -4, -2, 4, 6, 2, -2], [0, -5, 4, -1, -5, 2, -1, 1, -2, -3], [-12, 8, 4, -3, -6, -1, -6, 3, 9, 0], [-17, -3, -4, 1, 6, -8, 0, -2, 0, 3], [22, 6, 2, 2, 0, -6, -4, 0, 0, 2], [2, 10, -4, 0, 5, 0, 3, -2, 2, 2], [-9, -6, 0, 1, -6, 11, 2, 2, -3, -2], [-4, 5, -6, 0, -2, 5, 2, 6, -5, -5], [-3, -2, 2, -2, 2, 0, 5, 4, 2, 2], [-5, 0, -6, 2, 2, 0, 3, 4, 0, 0], [-8, 9, 0, 4, 3, 1, 0, 1, 1, -1], [2, -12, -8, 3, 2, 1, 0, 3, -5, -2], [3, -2, 0, 3, 1, -3, -3, 3, 4, 4], [-6, 4, -8, 1, 5, 11, 1, 0, -6, -4], [10, 0, -8, 0, 4, -4, -2, -2, 2, 4], [4, -2, 0, -1, -2, 7, 0, 1, 5, -2], [14, -3, 0, -4, 2, -5, -4, 0, 3, 3], [11, -1, 4, 1, 4, -6, 1, -7, -4, 1], [11, -5, 4, -4, 0, -3, -4, -3, 3, 3], [-3, -7, -2, 1, 0, 0, -2, -2, -8, -1], [-1, -6, 2, -3, 2, 9, 0, -2, -5, 2], [-2, 3, -10, 0, -1, -7, -1, 1, -2, 3], [-1, 7, 6, -1, -8, 2, 4, 0, 6, 1], [-5, 15, 0, 0, -2, -3, 2, 1, 1, 1], [-1, 5, -6, 2, -2, 3, 4, 3, -3, -3], [-1, -9, 12, 2, -2, -7, 0, 1, -1, -3], [13, -16, 10, 1, 6, -7, -2, 0, 9, 4], [23, 1, 6, -2, 0, -5, 0, -3, 5, -1], [-5, 9, -2, 0, 3, -1, -3, 3, 3, -1], [20, -6, -6, 0, 10, -6, 0, -2, -2, 2], [3, -18, -2, 1, 4, -9, -4, 0, 7, 6], [-13, -1, 4, 3, 0, -2, 0, -2, 0, 1], [9, 2, 0, -1, -2, 7, 6, -4, -3, -2], [-10, 0, 6, 2, -7, 2, 5, 4, 0, 0], [5, 7, 0, 0, 0, 5, 2, -1, -9, -3], [0, 14, 8, -1, -1, -3, -3, 3, 3, -2], [-3, 0, 0, -3, 2, 9, 2, -2, -5, 0], [4, 4, 4, -6, 1, 0, 3, -2, 2, 4], [5, 1, 6, -3, 2, -2, -6, -2, 2, 3], [-16, 10, 0, 7, -3, 5, 4, 5, 2, -4], [-9, 10, -2, -3, -2, 5, 6, -1, -2, -2], [-9, -2, 4, 0, 3, -4, -3, 3, 6, -2], [-4, 1, -2, 4, 3, -3, 0, 2, 2, -5], [-16, 10, 0, 0, 2, 0, -4, -6, -2, 2], [-20, 0, -4, 4, -2, 10, -2, 1, -9, -4], [-14, -3, -8, 1, -3, 8, 3, -4, -5, -1], [-24, -1, 0, -8, -1, 7, 0, -3, 1, 5], [-4, -4, 6, 0, -5, 4, 1, -5, -3, -6], [0, -7, 6, -1, -1, 4, -1, -4, 5, 3], [17, 0, 0, -2, -2, 6, 0, 7, 2, 2], [-10, 12, 2, 2, -8, 10, 4, 1, -7, -8], [-5, 5, 2, -2, -4, 1, 2, 7, 9, 3], [13, -21, 0, 2, 1, -5, 3, 0, 2, 1], [12, 9, 4, -1, -2, 0, 0, -1, 0, 5], [-3, 14, -10, -3, -2, 9, -2, -3, -10, -2], [-2, 1, -2, 0, 2, -7, -5, 0, 6, 3], [-6, -1, -4, 0, -3, -1, -2, -2, 8, 1], [11, -14, 0, 1, 0, -3, -2, -4, -5, 2], [10, 0, 2, -4, 0, -2, -4, -4, -2, 0], [-20, 1, 2, 1, -6, 8, 0, -3, -2, -5], [3, -2, 8, 3, 0, -3, 6, 5, 0, -6], [15, -11, 4, 1, 6, 4, 1, 5, -4, -1], [21, 0, 0, -2, 5, -4, -1, -4, -1, 0], [10, 6, -2, 0, 3, 0, 1, 5, 3, 0], [-12, 10, 4, 2, -4, 0, -4, -2, 0, 2], [-26, 1, 0, 6, -4, -1, 5, 7, 1, -1], [-16, 6, -10, 0, -6, 8, 2, 2, 0, -2], [2, 10, -8, -1, 1, 3, 3, 2, -2, -2], [13, 11, -4, 2, 4, -5, -2, -1, 1, 5], [6, 0, 0, -2, -6, 0, -4, 3, -3, 0], [-4, -4, -6, 4, -1, 2, 1, 0, -4, 0], [-13, -4, 4, -6, -2, -2, -7, -2, 6, 8], [-17, 12, -4, -2, -6, 4, -1, 6, 4, 0], [18, -15, 2, -2, 3, -3, 3, -3, 0, 1], [-6, -12, 6, -2, -4, 0, 2, 2, 2, -4], [-8, -6, 4, -4, 0, 2, 0, 0, 2, 2], [0, -3, 4, -5, 5, -4, -5, -2, -5, -1], [-23, 1, 2, 0, 3, -1, -7, -5, -7, -1], [3, -1, -4, 1, -2, -8, 2, 0, 6, 1], [-27, 9, 0, 4, 0, 1, 2, -1, -5, -7], [27, -5, 10, -3, -2, 4, -2, -2, 2, 1], [-13, 7, 4, 2, -4, 1, 0, -1, -5, -3], [1, -3, 14, -1, -6, 4, 2, -1, -1, -5], [-4, 1, 12, 5, -2, -2, 2, 0, -1, -1], [7, 11, -6, 0, 2, 1, 2, -4, -4, 1], [-18, 8, -4, 3, -1, -3, -1, 4, 2, 0], [-8, 3, -6, 0, 0, 9, 2, 0, 1, 1], [27, 1, 4, -3, 2, -2, 2, -2, 2, -1], [-22, -4, -12, 1, -6, 7, 4, -1, -5, -4], [-28, 4, -2, 2, 0, -4, -4, -8, -2, 0], [-10, 6, 0, 4, 0, 0, 2, 8, 4, -6], [0, 0, -6, 0, -8, 2, 0, 0, 4, -4], [2, -1, 0, -2, 0, -15, 1, -3, 3, 5], [6, -7, 12, -3, 5, -2, -7, -5, -4, 7], [-30, 3, 8, -1, -1, 2, -3, -3, 0, 1], [-9, 8, -4, 0, 0, 0, 5, 4, -10, 0], [-9, -7, 14, 4, 8, -9, -2, 3, 3, -1], [12, -10, -6, -2, 1, -4, -3, -2, 6, 6], [22, -6, 4, 0, 2, -8, -2, 1, 11, 10], [1, -9, -8, 1, -2, -10, -4, 0, 2, 1], [-3, -11, 4, -2, -3, 1, 5, -1, -3, -1], [-15, 0, 0, 3, 2, 1, -4, 0, -3, -4], [-12, 14, 8, 1, -4, -3, 0, -5, 3, 0], [6, -11, 4, -1, -4, -2, 2, -3, -2, 1], [18, 3, -2, 0, 1, -1, 1, 0, 9, 1], [14, -18, 8, -2, 6, -4, -8, -4, 4, 6], [-2, 5, 0, 1, 7, -10, -7, -6, -7, 3], [29, -5, 10, -1, 4, -2, 2, 2, -4, 1], [1, -4, 4, 0, -3, 6, 3, -2, 1, 0], [-24, 2, 4, 2, 12, 0, -2, 2, -2, -2], [15, -6, 10, -5, 0, -5, 2, 4, 1, 2], [-6, 9, -8, 0, 0, 5, 4, 6, 3, -1], [-29, 9, -8, 1, -5, 4, 4, 7, 2, -5], [-11, -11, 10, 0, -4, -1, 0, -1, 1, 7], [0, 1, -8, 3, -10, 4, -2, 3, 4, -1], [25, 3, 8, -2, -4, -11, -4, -1, 7, 3], [-1, -4, 0, 2, -2, 2, -6, -1, -12, -4], [10, -4, 2, -2, 10, -8, 4, -5, -3, -4], [-17, 12, 8, 2, 3, -6, 3, 2, -3, -4], [-19, -2, 2, 3, -9, 3, 1, 3, -2, -4], [8, -3, 2, -3, 0, 2, 4, 6, -1, 7], [11, -4, 8, -1, -2, 5, 6, -4, 1, -4], [33, -5, 4, -2, 5, -11, -9, -3, 7, 9], [-19, -1, -2, -1, -4, 6, 0, -6, 0, -3], [-8, -5, -8, 1, 7, 4, -5, -4, -3, 1], [-6, -5, -6, -1, 4, -4, -2, -4, 7, 5], [23, -4, 8, -2, 5, -4, 0, 1, 7, -2], [-20, -2, -4, 3, -6, 7, 4, 1, -7, -2], [-20, -4, 4, 2, -5, -4, -5, -4, 6, 0], [-1, 8, -2, -3, 8, 1, 0, -4, 1, 0], [-8, -8, -2, 6, 4, -16, 2, 3, 1, -4], [7, -12, -14, -3, 0, 3, 2, -4, -1, 4], [0, 2, 8, 4, 4, -6, 0, 0, 0, 2], [-19, 4, -4, 0, -12, 4, -1, 2, -2, 0], [-9, 7, 8, 0, 3, -7, 1, 0, 12, -3], [-13, -9, -6, -2, -6, 9, 6, 1, -1, -5], [-4, -2, -10, -4, -4, 14, 4, -4, -2, 2], [-3, -2, -4, 1, -5, 9, -1, 2, 1, -6], [17, -12, 4, -3, 2, -5, -2, -6, -3, 0], [0, -20, 4, 0, 12, -6, -4, -1, 1, 0], [7, -16, 6, -3, -3, -3, -1, 0, 3, 0], [13, -1, 2, 0, 10, -7, -4, -1, -3, 1], [12, -6, -4, 0, 0, 6, -2, 4, 2, -2], [7, -8, -8, -5, 2, 1, 2, 4, -1, 0], [-32, -8, -12, -2, 2, 6, 4, -2, -8, 0], [-15, 5, -10, 1, 2, 2, -2, 8, -2, -1], [-41, 15, -4, -2, -6, 3, 0, 4, 8, 1], [-13, -10, 0, -1, 2, 5, 4, -2, -1, 2], [-5, -7, -2, -6, 4, 7, -2, -3, 1, 1], [-39, 2, -10, 1, 6, 7, 0, -2, -7, 2], [-11, -17, -12, 4, 1, 1, 3, -4, -2, 1], [12, -15, 8, -1, 1, -10, -3, -1, 14, 3], [-28, 0, -12, 4, 8, 4, 6, -4, -10, 0], [-10, -2, 12, -2, -2, 0, 4, 2, 0, 2], [9, -15, -2, -2, -8, 11, 8, 1, -3, -1], [3, -5, 0, 1, -2, 6, 0, 2, -2, -3], [-19, -8, 2, 4, 2, -4, -9, -7, -1, 4], [-18, 10, -6, 3, 2, 3, -3, -2, -9, -2], [-1, 4, -8, 2, 0, 8, -3, -2, -8, -4], [11, -10, -10, 1, 13, -5, -1, -4, -5, -2], [26, -14, -4, 1, -1, -5, -5, 0, -2, -2], [21, 0, 2, -3, -2, -5, 8, -4, -1, 0], [-14, -9, -2, 1, -7, 6, 3, -9, -2, -3], [-37, 3, -2, 5, -2, 10, 4, 2, -8, -3], [-30, 0, -2, -2, 12, -4, -8, -4, -2, 4], [-27, 7, -2, 0, -2, -7, 0, -9, 1, -3], [17, -9, 6, -4, 0, -1, 0, -4, -2, 5], [17, -13, 4, -3, 10, -6, -8, -6, 4, 9], [-13, -1, -2, 1, -11, 14, 2, 2, -7, -11], [-15, 2, -4, 2, -6, 8, 10, 1, -2, -6], [10, 2, 16, -2, 6, -2, -4, 2, 10, 2], [15, -8, 2, 2, -4, -2, -5, -5, -5, 0], [10, -5, 8, -9, -2, -2, 0, 4, 7, 5], [20, 5, 8, -1, -6, 2, -2, -3, 6, 7], [-20, -8, -4, 2, 3, -2, -1, -9, -3, 2], [17, 1, -6, 1, 15, -16, -7, -2, 0, 3], [0, -2, 8, -6, -8, 4, -2, 0, 2, -2], [13, -9, 0, -2, 6, 5, 0, 6, -2, -3], [9, -16, 12, -3, 5, -7, -3, -1, 0, 2], [18, 6, 12, -1, 0, 3, 6, -1, -3, -2], [4, -8, 8, -3, 2, -1, -2, 7, 1, 4], [-27, 0, 4, 3, -6, 9, 2, 2, -1, -4], [19, -6, 12, -1, 0, -5, -4, -2, 11, 6], [2, -4, 4, 1, -4, -1, -2, 8, 2, -4], [-1, -3, 2, 1, -6, 0, 0, 6, 2, -5], [15, -14, 2, -2, 4, -14, -9, 0, 0, 6], [21, -5, 2, 4, 1, 3, -1, 4, -8, -3], [-39, 2, -14, -1, 2, 5, 0, 1, 0, 2], [-19, 15, 8, 4, 4, -11, -4, -7, -1, 5], [23, 2, -4, 0, -4, -6, 3, 2, 0, 2], [-8, 5, -8, 3, 2, -4, 0, -2, -7, 3], [4, 6, -8, 3, -9, 1, -1, -1, -1, -6], [-15, -3, -4, 1, -8, 16, 1, 7, -2, -5], [16, -8, 0, 0, 8, 6, -4, 2, 0, 0], [28, -2, 14, -4, 2, 0, -2, 5, 7, 6], [8, 12, -2, 0, -6, -8, 6, 0, 4, 0], [-10, -2, -10, 6, -4, 6, 2, 6, 0, -10], [14, -9, 8, -4, -8, -1, -3, 3, 3, 1], [-31, 5, -10, -1, -8, 14, 6, 6, 0, -9], [30, -12, 20, -4, 2, 0, -2, 0, 4, 4], [1, -5, -8, 1, 7, -2, -2, 8, -7, -3], [-6, -16, 0, -2, 5, 4, 3, -2, -8, -8], [-13, 15, -6, 5, 0, -8, 0, 1, 13, -3], [7, -19, -6, -2, 2, -3, 2, -5, 1, -7], [13, -9, -6, 4, 2, -3, 4, 1, 7, -1], [-20, 14, 14, 0, -4, 0, 2, 1, -3, -6], [43, -5, 12, 0, 10, -11, -8, -4, 2, 13], [6, 11, -4, -4, -2, -5, 3, 1, 5, 1], [-11, 2, 8, -3, -2, -3, 6, 6, 5, -2], [15, -17, 4, -2, 3, 3, -1, 3, -3, 1], [-24, -1, -4, 2, -5, -3, 2, 1, -1, 1], [16, -14, 12, 0, 12, -10, -4, -4, 4, 2], [-19, 15, -4, 4, -14, 5, 6, -3, 3, 1], [2, -12, 16, -5, -7, 7, 5, -4, 8, 4], [5, -2, 0, 2, -2, 4, 5, 6, 8, 2], [-14, 18, -4, 4, 2, -6, -6, 8, 4, 2], [32, 0, 4, 3, -1, 3, 1, 2, 8, 0], [0, -1, 0, -1, 6, 8, 4, -7, -6, -7], [-25, 10, -4, -2, -4, -2, 3, 2, -4, 2], [-49, 6, -6, -2, -10, 12, 7, -1, -1, -10], [21, -1, -6, -1, 11, 0, -3, -6, -2, 5], [-1, -4, 0, -5, 4, -11, -6, -1, 12, 12], [5, -14, 4, -3, 4, -9, 2, -6, 3, -2], [-5, 3, 0, -2, -8, -1, 6, 1, 9, -1], [4, -10, -14, 4, 3, -4, 5, 0, -4, 2], [12, -3, 6, 0, -2, -9, 0, 5, 8, -5]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2415_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2415_2_a_w();". // 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_2415_2_a_w(:prec:=10) chi := MakeCharacter_2415_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_2415_2_a_w();". // 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_2415_2_a_w( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2415_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-12, -89, 160, 244, -196, -143, 87, 30, -16, -2, 1]>,<11,R![13056, -2888, -22130, 12018, 5694, -4185, -223, 415, -31, -9, 1]>,<13,R![25024, 9688, -46448, -9078, 21592, -3002, -1811, 448, 23, -14, 1]>],Snew); return Vf; end function;