// Make newform 4018.2.a.bt in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4018_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_4018_2_a_bt();". // 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_4018_2_a_bt();". // 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, -204, 608, 231, -579, -109, 181, 19, -23, -1, 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], [4512, -20680, -26574, 17781, 15220, -4759, -2594, 513, 132, -19], [-16484, 56056, -41760, -33721, 29562, 4525, -5712, 235, 314, -39], [-124, 56056, -45032, -33721, 29562, 4525, -5712, 235, 314, -39], [-7936, 3108, 6310, 2603, 1161, -3244, -740, 725, 55, -42], [-4516, -35880, -3192, 30673, 7846, -11521, -2208, 1909, 142, -101], [-1788, -6120, -20264, 18748, 16381, -8003, -3334, 1238, 187, -61], [-256, 38256, 10732, -44154, -6401, 15345, 1058, -2048, -51, 91], [-19612, 1408, 92388, -30817, -53980, 18823, 10104, -3521, -552, 191]]; Rf_basisdens := [1, 1, 1636, 3272, 3272, 1636, 3272, 1636, 1636, 3272]; 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_4018_a();" function MakeCharacter_4018_a() N := 4018; order := 1; char_gens := [493, 785]; 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_4018_a_Hecke(Kf) return MakeCharacter_4018_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, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 1, 0, 0, 0], [0, 1, 0, -1, 0, 0, 1, 1, 0, 0], [2, 0, 0, 0, 0, 0, 0, -1, 0, -1], [1, 0, 0, 0, 0, 1, -1, 0, 0, 1], [0, 1, -1, -1, 0, 0, 1, 0, 0, -1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, -1, -1, 0, 0, 0, -1, 0, -1, -1], [2, -1, 1, 1, 0, 0, 0, 0, 2, 1], [3, -1, 0, 0, 2, 0, 1, -1, 0, 0], [2, 0, 0, 1, 0, 0, 0, 0, -1, 0], [0, -1, 0, 0, 0, -1, 0, 1, 1, 1], [0, 1, 0, -1, 0, 0, 0, 1, 0, 0], [0, -2, 0, -2, 2, 1, 0, 0, 0, 0], [-1, 1, -1, -3, 2, 0, 3, 2, -1, -1], [-1, 1, 1, -2, 2, -1, 3, 1, 0, 0], [-2, 0, -1, 0, 2, 1, 1, 1, -1, 0], [0, 0, 0, -1, 2, 1, -1, 0, -2, -1], [-1, 1, 1, 1, 0, -1, 0, -2, 0, 2], [-1, 2, 0, -1, 0, -2, 0, 0, 0, 0], [-2, 1, -1, 0, -2, -1, 0, 0, 0, -2], [3, 0, -1, 2, -2, 0, -2, 1, 1, 1], [1, 0, 1, -1, 0, 0, 0, 2, 0, 2], [7, 0, 1, 1, 1, 1, -1, -1, -1, 0], [1, 1, 1, -2, -2, -2, 1, 2, 0, 1], [1, -1, 0, -1, 4, 1, 0, -1, 0, -1], [4, 0, 1, 1, 2, -1, 1, -1, -2, 0], [-4, 3, 0, -1, 0, 0, 1, -1, 0, -2], [1, 1, -1, 0, -2, 2, -3, -1, 0, 0], [-4, 0, -1, -2, 2, 0, 1, 2, -2, -2], [4, 3, -1, -2, 2, -1, -2, -1, -2, 1], [4, 1, -1, 3, -2, -1, -2, 0, -1, 0], [1, 2, 0, 3, 0, 2, -1, -1, 0, -1], [4, 3, 1, 2, 0, -1, -2, -2, 1, 2], [1, -1, 0, -1, 0, 0, 0, 2, 2, 2], [-9, 4, -2, -3, 0, 0, 0, 2, 0, -2], [-1, 1, 1, -1, 2, -1, 1, -1, 2, -1], [-1, -4, -1, 1, -2, 0, 0, -1, -2, -1], [8, 0, 2, 2, 2, 2, 0, -2, -1, 2], [5, -1, -1, 2, 0, 0, -2, -1, -2, 0], [3, -4, 1, 1, 2, -1, 0, 0, 2, 0], [2, 0, 0, 4, -4, 2, -4, 2, 0, 0], [0, 2, -2, 0, 2, 1, 1, 3, 0, -1], [4, 2, -1, 3, -2, 1, -3, 0, 0, 1], [1, -1, 1, 0, 0, -4, 1, 0, 0, 1], [-3, 0, -2, -2, 0, -2, 2, 3, -2, -2], [12, 3, 0, 1, -2, -1, 0, 1, 3, 1], [0, -1, 1, 1, 2, 1, -2, 0, 1, 0], [-5, 4, -3, -1, -2, -2, 2, 5, 0, -1], [2, 0, 0, 3, 0, 3, 0, -1, 0, 0], [-2, 2, 1, -2, 2, 0, 0, -1, -1, -1], [-2, -1, -1, 0, -2, -1, 2, -2, 0, 0], [-1, -2, 1, 0, -1, 1, -1, 1, 2, 2], [2, 3, -3, 1, -2, 2, -1, 0, 2, -1], [2, 2, -2, -1, 0, 2, 0, 0, 1, 0], [-3, 1, -1, 0, -2, 0, -1, -1, 0, 2], [5, 1, -2, 3, -6, 2, -3, -3, 1, 0], [-3, 2, 0, 0, -2, -3, 2, 3, 0, 0], [-5, 3, 2, -1, 4, 1, 0, -1, 2, -1], [2, -2, 0, -1, -2, -3, 1, 0, 0, 1], [5, 1, 3, 5, 0, 1, -3, -4, 0, 2], [7, -1, 1, 1, 3, 0, -1, -2, 1, 1], [4, 1, 0, 2, -4, -2, -3, -2, -2, -2], [-4, -5, 0, 2, -2, -1, 0, 1, 0, 3], [2, -1, 0, 1, -3, -1, 0, -1, -1, 3], [-9, 0, -1, -2, 0, -1, 3, -1, -2, -2], [5, 1, 2, -3, 4, -1, 2, 3, 0, 1], [4, -4, 2, 2, -2, 0, 4, -2, 1, 0], [-8, 3, -2, -2, 4, 2, 1, 1, -1, 0], [0, -1, 2, -2, 6, 0, 4, 1, 2, 1], [1, 1, 3, -4, 0, -4, 3, -2, -1, 3], [-5, 1, 0, 2, 2, 2, 1, -1, -2, 0], [2, 1, 0, -2, -3, 0, -3, -1, -1, 0], [8, -2, 1, 2, -1, -1, -1, -3, 1, -2], [2, -5, 3, 0, 0, -4, 1, 0, 3, 3], [-1, 3, 5, 0, -6, -4, 1, 0, 0, 1], [-2, -1, 1, 2, -2, -1, -2, 0, 2, 2], [3, -3, 1, 2, -4, 4, -3, -2, 1, -1], [-4, -6, 1, -2, 0, 1, -2, 1, 0, 3], [-5, 5, 0, -5, 2, 0, 1, 3, 1, 0], [-4, 6, -1, 2, 0, -1, -3, -1, 0, 2], [5, -6, 0, 0, 0, 1, -3, 2, 0, -1], [11, 2, 2, 3, -3, 0, 0, -2, 0, 4], [-8, -3, -1, 4, -4, 2, -4, -2, 2, -2], [6, -3, 2, -1, 4, 2, -5, 1, 0, 2], [-11, 3, 1, 3, -8, 0, -3, 0, 0, 0], [-10, 5, -2, -4, -2, -1, 4, 3, 0, -3], [-2, -4, 1, 1, 4, 0, 6, -1, -1, -1], [-2, 5, -4, 1, 0, 0, -4, -1, 0, 2], [4, -1, 0, -1, 4, -1, -2, 0, -2, -1], [-2, 1, -1, -4, 6, 1, 6, -1, -2, -3], [-8, 4, -2, -1, 4, 1, -1, 0, -4, -1], [5, -2, 0, 2, -4, -5, 1, 0, 0, 3], [-2, 2, -1, -3, 2, -5, 9, 3, 0, 2], [2, -1, 2, 2, 2, -1, 5, 5, 2, -1], [1, 7, 1, -3, 0, -5, 2, 1, 4, -1], [-19, -1, -2, -2, 6, 5, -3, 2, 0, -1], [12, -2, 1, 6, -6, 1, -3, -1, -1, 2], [-2, -4, 0, 4, -4, 1, 0, -5, 0, 3], [5, -3, 0, 2, -2, 2, 1, -3, 0, -2], [12, -1, 0, -2, 6, 0, 5, 2, 0, 2], [0, -1, -1, 3, -4, 2, -5, -2, 0, 1], [6, -3, -2, -1, 4, 0, -1, 3, -2, -2], [8, 2, 3, 0, 2, 0, -2, 1, 2, 3], [-2, -4, 0, 3, -2, 3, -1, -2, 2, -3], [0, 3, 2, 1, -1, 2, -1, 1, 6, 4], [10, -5, 0, 1, 4, -3, 0, 1, 2, 1], [16, -1, 3, 8, -4, 2, -1, -4, 1, 3], [-7, -9, -1, 1, -4, 1, 0, 2, 0, 4], [8, -4, 1, 6, 2, 0, 0, 1, 2, 1], [-2, -1, -2, 2, -2, 0, 2, 0, 4, -2], [-15, -2, -1, -1, 2, -2, 2, -5, 2, 1], [14, -3, 3, 4, -2, -3, 2, -1, 2, 1], [-4, -4, 2, 2, 0, -2, -1, 0, 0, -2], [-2, 8, 2, -3, 1, 1, -1, 0, 4, -1], [8, 2, -1, -1, 0, 1, -7, -1, -2, 0], [-4, 1, -3, 0, -1, -1, -2, 0, 0, 0], [12, 4, -1, 2, -8, 0, -1, 1, -2, 1], [-6, 0, -1, 1, 4, 4, 1, 2, -2, -1], [-2, -4, -3, -2, -3, -4, 0, -5, -4, -1], [2, -2, 1, 6, -6, 0, -6, -6, 2, 0], [15, -8, 2, 2, -2, 0, 4, 2, 0, 1], [-13, 5, 1, -2, 4, 2, -2, 1, 0, -2], [5, 3, -2, -4, 2, 0, 1, 1, -6, -2], [0, 1, 0, -1, 2, -3, 2, -5, 1, -3], [3, 6, 0, 1, -2, 0, 0, 4, 2, 2], [-5, -6, 2, 1, 6, 3, -1, -4, -2, 2], [13, -5, 2, 2, 2, 0, 5, -4, 0, 1], [0, 1, 3, 3, 4, -2, 3, -2, 6, 3], [4, -8, 2, 1, 4, 1, -3, -2, 1, -1], [-6, -4, 0, 3, 2, 1, -1, 0, 4, -1], [-1, 3, -5, 1, -4, -1, -2, 4, -2, -2], [11, -3, 1, -2, 2, -6, 9, 2, 2, -1], [0, -5, 2, -2, -2, 3, -2, -3, 0, -1], [5, -2, 5, 4, 2, 4, -4, -1, -1, 1], [11, 2, 0, 4, -3, 3, -5, -2, 4, 1], [-11, -4, -1, 2, 4, 0, 0, -2, -4, -1], [1, 4, -1, 0, -4, -1, 0, 3, 4, 0], [2, 1, -1, -2, -2, -1, -2, 2, 4, 2], [-7, -4, 2, -1, 2, 3, 4, -2, 4, -6], [-5, 8, 4, 3, 0, 0, -4, -2, 3, 0], [-5, -6, -4, 4, 1, 4, -4, 0, -5, 0], [11, -8, -1, -4, 2, 2, 0, -1, -3, 1], [-13, 0, 0, 1, -2, 0, 2, -4, 2, 2], [3, -2, 1, -3, 4, 4, 0, 3, -1, 3], [12, 0, 3, -5, 6, 0, 4, 1, -2, 1], [-2, -1, -2, 3, 0, 2, -6, 1, -2, 0], [7, 2, -3, 3, 6, 5, -7, -5, -4, -3], [-1, 4, 0, -2, 6, -4, 6, 0, 1, 0], [-4, -4, -2, 5, -6, -3, -3, -2, -2, -3], [-21, 11, 1, -4, 2, -5, 2, 0, 0, -1], [17, -4, 1, 2, -2, 1, -1, -11, 0, -4], [-10, -2, 2, 3, -2, -5, 3, 0, -2, 3], [4, 8, -4, 3, 0, 5, -5, 0, 1, 1], [9, -4, 1, -1, -4, 3, 1, 3, 1, 2], [3, 9, -4, 4, 0, 1, -6, 1, -1, 1], [4, 3, 2, 1, 0, 1, 4, -1, 0, 4], [-9, -3, 3, -1, 8, 1, 3, -2, 4, 0], [2, -6, -1, -3, 0, 1, 1, 1, -2, 2], [-1, -2, -1, 0, 4, -3, 3, -3, -2, -2], [-16, -1, -1, 6, 0, 2, -5, -8, 0, 0], [7, 1, -1, 0, 2, 4, -5, -5, -4, -6], [5, 0, 3, -2, -2, 0, -2, -1, -1, 1], [-15, -2, -3, -5, 0, -6, 8, 3, 0, -1], [8, -2, -5, -4, 2, 3, 0, -2, 0, -2], [6, 6, -8, -1, -2, -1, -3, 0, -2, -9], [22, -3, -3, 2, 0, 2, 0, -3, 0, -3], [-10, -2, 1, -1, 6, 4, 0, 5, 4, 3], [-14, -5, -4, -4, -4, -4, 4, 4, 2, 0], [5, -13, 0, 7, -4, 1, 0, -3, -4, 1], [-5, 2, -3, 0, 6, 4, -2, -2, -2, -1], [4, -4, 0, -3, 0, -1, -3, -6, -3, -3], [-2, 2, 3, 4, -10, 1, -7, 5, 2, 7], [2, -5, -1, 2, 4, 3, -6, -4, -2, 2], [-7, 2, 4, -2, -4, -1, 3, 2, 2, 3], [24, 7, 1, 2, 0, 4, -5, -2, -1, 1], [3, -6, -2, 2, -2, -1, 3, -1, 0, -2], [-7, 6, 3, -8, 2, -4, 4, 3, 3, 3], [10, 4, -3, -2, 8, 4, -2, 1, -4, -3], [-7, -8, 3, 6, -6, -1, 3, 1, 5, 0], [6, 9, -1, 2, -2, -5, 4, 2, 2, 2], [9, -1, -3, -1, 0, -3, -2, -4, -2, -2], [3, 1, 1, 7, -4, 1, -6, -10, 2, 4], [-1, 3, 4, 10, -10, -1, -7, -4, 2, 1], [3, -6, -2, -2, 2, -6, 2, -2, -3, -4], [1, 5, 2, 0, -2, -1, -5, -2, -4, 1], [3, -5, 2, -5, 6, 2, -5, 1, -1, 0], [-19, -1, 1, 0, -1, 2, 1, 6, 2, 1], [17, -1, 0, -2, -4, -1, 2, -1, -3, -3], [0, 6, -3, 0, -6, -1, -5, 3, -5, 0], [11, -9, 3, -4, 6, 2, -1, 2, 4, 5], [2, -6, 4, 7, -4, 5, -3, 4, 2, 1], [3, 5, -1, -3, -8, 1, -6, 6, -4, -4], [30, 2, -2, 7, -2, 2, -3, 0, 0, 1], [18, 6, 1, 7, -11, -1, -3, 1, 2, 4], [-9, 2, -3, -7, 6, -2, 6, 3, -2, -3], [-1, 5, 2, 3, 4, 5, 2, -2, 2, 0], [-17, 4, -4, -2, 0, -5, 1, 0, 2, -7], [0, -12, 4, 1, 2, 1, 2, 0, -2, 3], [3, 3, 3, -7, 2, -5, 6, 3, 0, 1], [3, 2, -3, 0, -4, 1, -3, -1, -8, -6], [4, -9, 0, -1, -2, 1, 7, 2, 0, 1], [4, -4, 5, 0, 8, -5, 3, -1, 6, 4], [-16, 10, -4, -11, 4, -8, 10, 8, -3, 0], [6, -2, 2, -6, 6, 2, 4, -2, 4, -2], [-19, 10, -4, -2, -6, -1, -1, -1, -2, -6], [-8, -12, -1, -1, 2, 3, 3, -3, -4, 0], [21, 11, 5, 5, -4, 1, -10, -6, 2, 6], [-12, 3, -5, -10, -2, 0, 9, 4, -2, -4], [-14, -3, 0, -5, 12, 1, 8, 5, 0, -1], [5, 11, -2, 0, 2, -3, 0, 3, 0, 0], [8, 3, 3, -3, 8, 2, 1, 0, 2, -1], [16, -5, -3, 5, -2, 2, -6, -9, -2, -6], [1, -6, 0, -7, 6, -4, 4, 12, -4, -2], [6, 5, -4, -2, 4, 5, 2, -1, 0, 1], [1, -9, -6, 2, -2, 2, 3, 1, 0, 0], [6, 12, -2, -5, 0, -3, 7, 2, 0, 1], [-9, -2, -8, 2, -6, -2, -8, 0, -6, -3], [-11, 15, -4, -8, -4, -7, 10, 11, -1, -3], [8, 3, 2, 0, 0, -5, 6, -1, -4, -3], [-5, -6, 0, 4, -9, -6, 2, 2, -3, 0], [-7, -6, -2, 5, -10, 0, -8, -6, 0, 4], [-10, 12, -2, 0, -2, 1, -1, 7, 4, 1], [9, 7, -3, -5, 2, -2, 1, 2, -5, 3], [-9, 2, 2, 0, -13, -3, -5, 0, 2, -3], [-21, 4, 0, -1, -6, -2, 5, 0, 4, -2], [-17, -1, 0, 6, -6, 2, -9, -5, 0, 0], [9, -3, 1, -3, -7, 0, 3, -4, -3, 1], [-12, -1, -2, -3, 8, 1, 1, 2, -2, -5], [-4, -1, -2, 0, -8, 4, -3, 5, 4, 3], [8, -5, -2, 5, 6, 4, -7, -1, -4, 0], [11, 11, -3, 1, 2, 5, 0, -1, 2, -1], [-9, 0, -7, 3, 0, -1, 3, 9, -5, 0], [5, 11, 3, -5, 2, 1, 1, 3, 4, 3], [0, 0, -1, 0, 8, 8, -7, 1, 0, -3], [-6, 2, -7, -1, -2, -4, -2, 2, -4, -3], [15, 7, 5, 8, -4, 3, -6, -4, 0, 5], [20, -13, 4, 5, -4, 4, -3, -3, 6, -2], [35, -8, 2, 2, 0, -3, -3, -4, 2, 3], [-31, -3, -5, -4, -2, -4, 3, 10, -2, -3], [-18, 2, -1, -4, -8, -6, 0, 1, 0, -5], [17, 1, -1, 8, 4, 5, -7, 3, 0, 0], [-11, -7, 0, -7, -2, -2, 7, 9, -1, -4], [7, -4, 7, 2, 4, 3, -3, -7, 6, 0], [-4, 3, -4, -7, 2, -4, 1, 1, -3, -4], [3, -3, 5, 5, 0, 0, -5, -8, 4, 3], [-1, 7, -5, -2, 2, -8, 3, 0, 2, -5], [0, 1, 0, 4, -1, -3, 4, -1, 2, 1], [23, -3, -5, -4, -2, 4, -1, -1, -4, -6], [-5, 6, 0, 5, -8, -1, -1, -2, 5, 1], [11, 6, -1, -1, 0, 10, -4, 0, -6, 0], [1, 2, 2, 4, 0, 1, -10, -6, -4, -1], [-4, -5, 0, 10, 2, 7, -2, -6, 0, -4], [-7, -1, 0, 1, -6, -5, -2, 5, 2, 3], [-11, 0, 0, -2, 4, -3, 1, -2, 2, 3], [12, 3, 3, 4, -10, 0, -3, -1, 0, 1], [1, -4, -2, 8, 4, 1, -5, 1, -2, 4], [-10, -10, -2, -9, 8, -3, 9, 11, -4, -4], [12, -10, 4, -2, 4, -7, 7, -4, -8, 1], [3, -7, -4, -5, 6, -2, -1, -3, -3, -2], [-7, -6, -3, 4, 2, 1, -7, -4, -6, -3], [6, 6, 0, 1, -6, 4, -8, -4, 0, -2], [0, 9, 0, 0, 4, -2, 3, 3, -3, 2], [8, 7, 2, 2, -8, 3, -2, -2, 0, 8], [-23, 3, -2, -3, 0, -3, 4, -2, -4, -2], [10, -4, 1, -5, 4, 8, -4, 3, -1, 5], [10, -4, 1, 3, 3, -1, -5, -1, 4, 0], [-12, 6, -4, 7, 2, 5, -5, -10, 2, -5], [15, 6, -4, -11, 14, 2, 4, 2, -4, -2], [9, -7, -1, -9, 8, 3, 0, 2, -4, -2], [12, 5, -2, -4, -2, -1, -1, 7, 0, -1], [9, 7, 0, 6, 6, 4, -7, -2, -6, 3], [20, -9, 2, 2, 2, -7, 4, 1, -6, 1], [-15, 5, -5, -6, 0, -5, 4, -3, -6, -6], [-3, 0, 1, 5, 0, 6, -7, 0, 4, -2], [-14, 12, 4, -4, 0, 1, 3, 1, 6, -1], [19, 1, -4, 3, -8, -3, -4, 3, 0, 5], [16, 6, -1, -2, 3, -8, 4, 3, 4, 1], [-3, -6, 1, -2, 4, 3, 0, 3, 4, 2], [5, -5, 7, -3, 2, -4, -3, 1, 0, 7], [14, -5, -7, 0, 6, -3, 0, 2, -4, -2], [22, -13, 3, 2, -1, 9, -6, -6, 0, -2], [1, 3, -6, -9, 4, -5, 15, 11, -2, 1], [-7, -10, -2, 2, -8, -9, 3, -2, -2, 1], [-2, 13, 5, -4, 6, 1, 6, 2, 4, 0], [4, 10, -1, 4, -4, -4, 0, -1, 4, -1], [8, 4, -6, 6, -10, -4, -4, 6, -4, 2], [-11, -3, 6, -1, 2, 0, 1, 3, 3, 4], [17, -3, 4, 6, -6, -2, 1, -4, 6, 5], [-10, -7, 3, 6, -2, 1, -6, 1, 2, -1], [-26, 8, 7, 7, -2, 0, -3, -2, 6, -1], [6, 2, 4, 3, -8, 1, -5, -6, 3, 5], [2, -3, 1, 6, -10, -4, -5, -11, 4, -1], [-22, -8, -1, 10, -4, 3, 1, 0, 0, -6], [16, -7, -1, 0, 3, 10, -5, -2, -3, 5], [2, -5, 4, -6, 6, -5, 2, -6, 2, 4], [8, -10, 1, -3, 10, 3, 1, -5, -6, -2], [5, -4, 0, -7, 10, 2, -1, -2, 0, 0], [-16, -3, 4, 1, 4, -2, 9, 3, 0, -8], [22, 8, 3, 0, 8, -2, 2, -3, 2, -1], [16, 3, -4, 3, 4, 6, -5, -7, -4, -4], [15, 9, 0, 2, -1, 0, 5, 3, 2, 2], [22, 8, 1, -3, 0, 6, -6, 1, -6, 0], [-7, 3, 1, 0, -4, -2, -1, 0, 6, 3], [-15, -2, 1, -4, -6, 3, 10, 2, 2, -3], [3, 0, -5, 9, -6, 2, 2, -3, 4, -1], [15, -7, 5, 2, 4, -2, -3, 0, 1, 7], [0, 5, 2, 4, 4, -6, -3, -3, 1, 4], [23, 3, 2, 17, -12, -3, -8, -9, -2, 1], [22, -7, -2, -4, 5, -2, -1, 3, -1, -2], [28, -1, 7, -7, 8, 2, 3, -2, -2, 5], [10, 7, 6, 0, -6, 1, 0, 7, 2, 3], [4, -8, -2, 2, 0, 0, 8, -8, -2, -8], [12, 1, 6, 0, 8, 3, 0, -1, 0, 7], [25, -7, 4, 11, -2, 2, -3, 5, 1, 4], [-13, -12, -5, 3, -8, 2, -6, 3, 3, 3], [-3, -11, 0, -7, -2, 1, 4, 5, -1, 1], [-26, 6, 1, -3, -4, -2, 5, -4, 6, 1], [11, -1, -4, 6, -4, 1, -10, -13, -3, 1], [-25, -2, 2, 8, -8, 1, -3, -2, -2, -1], [12, 2, 5, 10, -8, 3, -10, -6, 2, 8], [-6, 2, 3, 8, -8, 5, -11, -2, -4, 6], [27, -18, 3, 2, -4, 2, -3, -7, 0, -6], [18, -1, -2, 5, 3, 2, 5, 7, 0, 2], [29, 5, 0, -7, 4, 1, 2, 2, -4, 0], [37, 2, 6, 3, 2, -2, -2, -10, 0, 6], [20, 0, 1, -1, 2, -7, 5, -3, 0, 0], [-16, 6, 4, 9, -6, 6, -3, -11, 6, 2], [4, 11, -6, 3, -4, -2, 3, -5, 2, -4], [4, -3, 5, 4, -10, 3, -3, -8, 0, 0], [11, -3, 1, 1, 10, 7, -4, -5, 2, -1], [22, -3, -4, 4, 4, -2, -5, 3, -7, 6], [5, -7, -4, 4, -10, 8, -9, 3, 4, -2], [2, -6, -1, -1, -2, -1, -1, -9, -2, -6], [-30, 1, -2, -1, 6, 0, 3, -2, -2, -11], [10, 2, -3, 1, 0, 5, -15, -5, -1, 10], [13, -10, -5, -1, -2, 8, -2, -1, -8, -3], [-31, 9, -4, -9, 6, -4, 7, 11, -4, 1], [-10, 2, 1, 1, -2, -7, 9, 11, 0, -4], [16, -4, -4, -4, 10, 7, -9, 0, -9, 1], [0, 1, -2, -7, 7, 8, 5, 5, 2, -4], [24, -2, 3, -4, 4, 0, -2, 3, 4, 7], [41, -1, -3, 2, 0, 0, 0, -7, -2, -6], [18, -6, 4, -6, 0, -4, 0, 0, 6, -2], [4, 19, -1, -2, -6, 1, -3, 0, -6, 0], [-13, 10, -3, -4, -10, -4, 2, -1, -1, -5], [-5, 7, -1, -11, 16, 5, 0, 3, 2, -5], [13, -5, 4, 5, 0, -7, 0, -1, -6, -1], [2, -15, 0, -4, 2, 7, 0, 0, -6, 8], [-36, 6, -7, -11, 4, -1, 7, 7, -4, -10], [5, -5, -3, 3, 4, 3, -4, -6, 0, -2], [-11, 1, 0, -1, -6, 0, 5, 7, 1, -2], [36, -9, 0, -3, 6, -5, 2, -1, 1, 3], [6, 1, -7, -4, -14, -5, 6, 0, 0, -2], [-5, 4, -1, -7, -4, 2, -2, 5, -3, 1], [-19, 3, 4, -4, -10, -2, 7, -5, 2, 2], [29, 4, 6, 2, -2, -2, 10, 3, -2, 0], [11, 3, -8, -1, 4, -13, 4, -1, -2, -5], [0, 9, 0, -4, -10, -7, -3, -1, 2, -3], [-11, -8, 4, -3, -2, 8, -4, -4, -4, -4], [-18, 10, -2, 10, -8, 2, -4, -5, -2, -1], [19, -3, 5, 8, 2, 2, -3, -6, 4, 9], [18, 17, 5, 4, -4, 5, -6, -2, 7, 10], [-28, -18, -3, -1, -10, -4, 3, 6, 2, 5], [-7, 6, -2, -7, -2, -1, 4, 11, 0, -5], [18, -4, 4, 0, -8, 2, -4, -8, 4, -2], [3, 4, 6, -11, -2, 4, -4, -2, 2, -2], [5, 7, 3, -1, 4, 1, -10, -4, -2, -2], [-13, 11, -2, -11, 6, -5, 2, 8, 2, 2], [-26, -1, 2, -11, 10, -3, 8, 9, 5, 3], [-5, 10, -4, 3, 6, 4, -2, 0, -4, -2], [7, -6, 10, -3, 4, -4, 2, -2, 3, 2], [2, 7, -2, 0, -2, -7, 4, 3, -2, -1], [-2, -14, -6, 6, -4, -6, -4, -8, -8, -4], [-18, 3, 3, 2, -12, -3, 8, -6, 0, -2], [20, 1, -2, 0, 2, 6, -3, 1, -2, 2], [-13, -1, -2, 2, 2, 6, -15, -7, -2, 0], [8, 9, 0, -6, -5, -1, -4, -1, 6, -1], [-21, 2, -3, -7, 2, -7, -3, -10, -8, -4], [22, -1, 2, -1, 8, -8, 3, -3, 0, 6], [-31, 4, 3, 5, 6, 7, -4, -1, -2, -1], [-3, 13, 1, 4, 0, -4, 3, -2, 2, 1], [1, -4, 9, -2, 10, 4, -6, -7, 1, -1], [34, -8, 5, 9, 2, 0, 5, -2, 4, -1], [11, 6, 0, -6, 2, -5, 7, -6, 0, 1], [9, 1, -1, 0, 8, -12, 7, -5, 0, -2], [1, -13, 7, 1, 6, 2, 6, -1, 4, 1], [2, 0, 2, 7, 0, -4, 8, -8, 1, 4], [1, -6, 4, 11, -14, -2, -4, -8, -4, 4], [-30, 14, -4, 6, -6, 0, -2, 6, 2, 0], [-11, 0, -2, 5, -8, 5, -2, -2, 6, 4], [4, -12, -3, -5, -11, -9, -3, 1, -2, -2], [1, 11, 1, 1, -12, -5, -4, 8, 0, 2], [-14, 6, -6, -15, -6, -5, 7, 6, -4, 1], [-1, -1, -2, 9, -19, -1, -12, -3, 2, -1], [-4, 1, 0, -13, 8, 2, 1, 3, 2, 0], [-29, 2, -10, 2, 4, 1, -1, 12, 0, 1], [-3, -11, 2, -2, 10, 6, 1, -1, -4, 4], [8, -8, -6, 9, -6, -5, 1, -4, -2, 5], [2, 7, 5, -5, 10, -2, 4, 3, -2, 6], [-26, -5, 2, -11, 15, 4, -3, -1, 2, 0], [-14, 7, -6, 8, -4, 1, -8, -5, 0, 3], [-46, -3, -2, 0, -4, 2, 2, 6, 4, 0], [3, 4, 3, 0, 0, 1, -9, -7, -2, 2], [-10, 1, 0, 1, 8, 9, 0, -5, 0, 1], [2, -14, 1, 1, 13, 3, -5, -1, -4, -2], [-6, -3, -3, -10, 4, 0, 15, 3, 2, -3], [-4, -10, -1, 2, 2, 7, 0, 0, 10, -4], [-18, -3, -3, -2, 6, -9, 8, 0, -2, -4], [20, -4, -2, 6, -8, 0, -4, -2, 0, 2], [9, -7, -1, -4, -6, -6, 17, 0, -2, -5], [10, 4, -2, -5, 12, -2, 3, 10, 0, 9], [11, 5, 3, 0, 4, 8, 3, 6, 2, 1], [-26, -9, 1, 2, 6, 5, 4, 6, 0, 2], [15, -12, -11, -1, 10, 2, 2, 3, -6, -3], [-3, 10, -2, -7, -4, 0, 6, -5, -2, 1], [-5, 2, -5, 0, -8, -5, -9, -3, -6, -6], [22, -9, 10, 0, 10, -2, 0, -6, 4, 2], [4, -1, -6, 5, -11, 1, -10, -9, -5, 1], [-25, -2, -2, 8, 2, 1, 1, -10, 7, 1], [21, 6, -7, 8, 4, 0, -1, 4, 0, -1], [-3, 3, -7, -10, -8, -10, 11, 0, -2, -1], [10, 5, -1, 11, -4, 9, -2, -1, -4, 2], [8, 5, 6, 6, 8, 8, -4, -5, 4, 7]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4018_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4018_2_a_bt();". // 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_4018_2_a_bt(:prec:=10) chi := MakeCharacter_4018_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_4018_2_a_bt();". // 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_4018_2_a_bt( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4018_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-12, 204, 608, -231, -579, 109, 181, -19, -23, 1, 1]>,<5,R![84, -740, -742, 2963, -1137, -808, 373, 71, -35, -2, 1]>,<11,R![-2268, 38592, -54918, 8589, 14070, -4429, -845, 415, -6, -11, 1]>],Snew); return Vf; end function;