// Make newform 5625.2.a.y in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_5625_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_5625_2_a_y();". // 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_5625_2_a_y();". // 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 := [145, 0, -180, 0, 80, 0, -15, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-4, 0, 1, 0, 0, 0, 0, 0], [14, 0, -8, 0, 1, 0, 0, 0], [0, 48, 0, -44, 0, 12, 0, -1], [-53, 0, 45, 0, -12, 0, 1, 0], [0, -53, 0, 45, 0, -12, 0, 1], [0, 66, 0, -53, 0, 13, 0, -1]]; Rf_basisdens := [1, 1, 1, 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_5625_a();" function MakeCharacter_5625_a() N := 5625; order := 1; char_gens := [4376, 1252]; 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_5625_a_Hecke(Kf) return MakeCharacter_5625_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, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 2, 0, 0], [0, -1, 0, 0, -1, 0, -1, 1], [-2, 0, 0, -1, 0, -2, 0, 0], [0, 2, 0, 0, 0, 0, 0, 1], [0, 0, -2, -1, 0, 0, 0, 0], [0, -1, 0, 0, 1, 0, 0, -1], [0, -3, 0, 0, -2, 0, 0, -1], [-1, 0, -3, 3, 0, 0, 0, 0], [0, 0, 3, 3, 0, 0, 0, 0], [0, -1, 0, 0, 1, 0, 1, 1], [-7, 0, -1, 2, 0, 0, 0, 0], [0, -2, 0, 0, 1, 0, -2, -3], [0, 0, 0, 0, -1, 0, 1, -2], [0, -1, 0, 0, 2, 0, 2, 1], [5, 0, 3, 0, 0, -3, 0, 0], [-5, 0, 4, 2, 0, -1, 0, 0], [0, -3, 0, 0, 2, 0, -2, -4], [-6, 0, 3, -2, 0, -2, 0, 0], [0, 0, 4, 0, 0, 1, 0, 0], [0, 3, 0, 0, 2, 0, 0, 4], [0, 4, 0, 0, -2, 0, 3, 1], [-10, 0, -2, 5, 0, -4, 0, 0], [0, 4, 0, 0, 3, 0, -1, -1], [-1, 0, -3, -5, 0, 7, 0, 0], [0, -1, 0, 0, -2, 0, 2, 0], [-1, 0, 1, 0, 0, -2, 0, 0], [0, -5, 0, 0, -3, 0, 1, -1], [-7, 0, 0, -4, 0, -1, 0, 0], [0, -5, 0, 0, 0, 0, -6, 2], [0, 1, 0, 0, 0, 0, 2, 5], [-3, 0, 6, -1, 0, 5, 0, 0], [0, 2, 0, 0, 4, 0, 8, 0], [10, 0, -3, -5, 0, 8, 0, 0], [-2, 0, -4, 2, 0, 4, 0, 0], [-8, 0, -4, 7, 0, -3, 0, 0], [0, -3, 0, 0, -3, 0, -4, -1], [0, 3, 0, 0, 4, 0, 1, -2], [0, 3, 0, 0, -1, 0, 8, -1], [8, 0, -2, -8, 0, 12, 0, 0], [0, 2, 0, 0, 5, 0, 4, 0], [-3, 0, 9, -2, 0, -9, 0, 0], [0, -2, 0, 0, -2, 0, -3, -3], [11, 0, -4, 5, 0, 7, 0, 0], [-5, 0, -10, 1, 0, -1, 0, 0], [-5, 0, -1, 2, 0, 4, 0, 0], [0, 8, 0, 0, 2, 0, 0, 3], [1, 0, -6, -10, 0, 2, 0, 0], [0, 0, 0, 0, 2, 0, 0, -1], [0, 2, 0, 0, 3, 0, 2, -5], [-3, 0, 8, -3, 0, -13, 0, 0], [0, 1, 0, 0, 0, 0, -6, 1], [0, -5, 0, 0, -3, 0, -2, -1], [0, 3, 0, 0, -2, 0, 3, 3], [0, 7, 0, 0, 0, 0, -3, -1], [-14, 0, -1, -1, 0, -2, 0, 0], [-6, 0, 9, 2, 0, -13, 0, 0], [0, 7, 0, 0, 2, 0, -1, 4], [9, 0, -5, -15, 0, 8, 0, 0], [0, -5, 0, 0, 1, 0, -2, -5], [-8, 0, 4, -4, 0, 0, 0, 0], [0, -7, 0, 0, -5, 0, -3, -7], [4, 0, -4, 10, 0, 10, 0, 0], [0, 8, 0, 0, 2, 0, 6, -1], [13, 0, 6, 6, 0, -3, 0, 0], [12, 0, 5, -11, 0, 6, 0, 0], [0, -9, 0, 0, 1, 0, 0, 2], [7, 0, -7, -2, 0, -9, 0, 0], [0, -1, 0, 0, 4, 0, -4, -4], [0, 0, 0, 0, -9, 0, 1, 5], [0, 0, 5, 3, 0, -16, 0, 0], [0, 0, 3, 11, 0, -16, 0, 0], [2, 0, -9, -13, 0, 19, 0, 0], [0, -5, 0, 0, -11, 0, -3, 5], [0, -2, 0, 0, 2, 0, -5, -4], [-9, 0, -1, 16, 0, -12, 0, 0], [0, 4, 0, 0, -1, 0, 9, 1], [-12, 0, -2, 4, 0, -19, 0, 0], [0, 6, 0, 0, 3, 0, 4, -3], [-13, 0, -5, 11, 0, -4, 0, 0], [0, 3, 0, 0, -3, 0, -3, -3], [2, 0, 0, 3, 0, -12, 0, 0], [-5, 0, 5, -5, 0, -12, 0, 0], [0, -6, 0, 0, 0, 0, -3, 0], [0, 2, 0, 0, -1, 0, 0, -9], [-2, 0, -8, -14, 0, 13, 0, 0], [0, 2, 0, 0, 0, 0, 1, -9], [-24, 0, 7, 0, 0, -9, 0, 0], [0, 7, 0, 0, 6, 0, 12, 2], [0, 4, 0, 0, 4, 0, 0, 4], [-5, 0, -4, 7, 0, 8, 0, 0], [0, 15, 0, 0, 5, 0, 1, 3], [0, 0, -7, 3, 0, 23, 0, 0], [0, -7, 0, 0, 1, 0, -9, 2], [0, 0, 0, 0, -4, 0, -5, -3], [0, 1, 0, 0, -9, 0, -8, 2], [-8, 0, -14, 0, 0, 1, 0, 0], [-20, 0, 2, 5, 0, -5, 0, 0], [-31, 0, 4, -4, 0, -6, 0, 0], [0, 7, 0, 0, -5, 0, -6, -5], [0, 2, 0, 0, 0, 0, 2, 13], [0, 9, 0, 0, 2, 0, -3, 1], [18, 0, 1, 2, 0, 15, 0, 0], [10, 0, -12, -1, 0, 13, 0, 0], [0, -2, 0, 0, 6, 0, -5, -4], [0, -6, 0, 0, -2, 0, 4, 1], [0, -2, 0, 0, -4, 0, -1, 6], [-7, 0, 0, -11, 0, 2, 0, 0], [-5, 0, -12, 7, 0, 7, 0, 0], [-9, 0, -4, -6, 0, -4, 0, 0], [0, -6, 0, 0, -6, 0, -8, -3], [8, 0, -13, -3, 0, 19, 0, 0], [-26, 0, -4, 12, 0, -3, 0, 0], [0, 8, 0, 0, 5, 0, 4, -6], [-9, 0, 16, 0, 0, -1, 0, 0], [0, -5, 0, 0, -9, 0, 5, -3], [0, 11, 0, 0, 8, 0, -4, 6], [0, 11, 0, 0, -5, 0, -4, 2], [28, 0, 3, -20, 0, 16, 0, 0], [-18, 0, -1, -7, 0, -14, 0, 0], [0, 3, 0, 0, -2, 0, -1, 1], [0, 1, 0, 0, 3, 0, 0, -11], [13, 0, -2, -5, 0, 26, 0, 0], [0, 12, 0, 0, 0, 0, 1, 6], [-9, 0, 11, -10, 0, -14, 0, 0], [0, 13, 0, 0, 1, 0, -2, -2], [-34, 0, -6, -3, 0, 6, 0, 0], [-24, 0, 8, -4, 0, -14, 0, 0], [10, 0, 15, 2, 0, -2, 0, 0], [0, -8, 0, 0, -2, 0, 4, -9], [-3, 0, 14, -1, 0, -7, 0, 0], [8, 0, -9, 8, 0, 17, 0, 0], [0, -3, 0, 0, -1, 0, 2, -13], [7, 0, 2, -14, 0, 9, 0, 0], [0, 2, 0, 0, -1, 0, 15, -2], [9, 0, 7, -1, 0, 19, 0, 0], [0, -14, 0, 0, 6, 0, 1, -4], [0, 9, 0, 0, -5, 0, 6, 4], [9, 0, 10, -9, 0, -5, 0, 0], [0, -6, 0, 0, -1, 0, -2, -7], [1, 0, 10, 4, 0, -16, 0, 0], [0, 0, 0, 0, 9, 0, 3, 3], [23, 0, -10, 13, 0, 12, 0, 0], [0, -4, 0, 0, -2, 0, -8, -4], [-16, 0, -2, 19, 0, -19, 0, 0], [0, 11, 0, 0, 2, 0, -1, 4], [-21, 0, -7, 11, 0, 5, 0, 0], [0, 1, 0, 0, -3, 0, 7, -1], [9, 0, -16, -2, 0, 15, 0, 0], [0, 6, 0, 0, -7, 0, -12, -1], [-28, 0, 4, 9, 0, -26, 0, 0], [0, -1, 0, 0, 6, 0, 10, -3], [18, 0, -3, -18, 0, 6, 0, 0], [0, -6, 0, 0, -7, 0, -8, 3], [-8, 0, 6, 9, 0, -5, 0, 0], [0, -2, 0, 0, 0, 0, 0, 2], [9, 0, -8, -8, 0, 23, 0, 0], [0, -4, 0, 0, 7, 0, 8, 7], [0, -3, 0, 0, -6, 0, -12, -3], [0, 2, 0, 0, -2, 0, 16, -2], [-4, 0, -8, 10, 0, 6, 0, 0], [0, 15, 0, 0, 3, 0, 3, 0], [0, 14, 0, 0, -5, 0, 4, 1], [0, 6, 0, 0, 4, 0, 3, 8], [1, 0, -7, -20, 0, -6, 0, 0], [12, 0, 5, 17, 0, -5, 0, 0], [7, 0, -5, 3, 0, -4, 0, 0], [0, 10, 0, 0, 0, 0, 8, 8], [0, 4, 0, 0, 3, 0, 6, 0], [-18, 0, 9, 7, 0, -11, 0, 0], [0, 5, 0, 0, 12, 0, 14, 1], [20, 0, 12, -2, 0, -17, 0, 0], [-11, 0, 12, 7, 0, 5, 0, 0], [0, -16, 0, 0, -6, 0, -3, 10], [26, 0, 19, -12, 0, -2, 0, 0], [0, -5, 0, 0, 0, 0, -13, 11], [-34, 0, 5, 17, 0, -17, 0, 0], [20, 0, -7, 18, 0, 23, 0, 0], [-1, 0, -20, -9, 0, 31, 0, 0], [0, -12, 0, 0, -5, 0, -2, 1], [37, 0, -6, -10, 0, 2, 0, 0], [0, -11, 0, 0, -2, 0, -7, -9], [0, -6, 0, 0, -9, 0, -10, 4], [0, -17, 0, 0, 2, 0, 0, 1], [20, 0, 1, -1, 0, 9, 0, 0], [35, 0, -6, -4, 0, 16, 0, 0], [5, 0, -22, 13, 0, 22, 0, 0], [0, 3, 0, 0, 8, 0, -6, 5], [-28, 0, 11, -2, 0, -20, 0, 0], [0, -10, 0, 0, 0, 0, -11, -10], [12, 0, -19, -14, 0, 38, 0, 0], [0, 0, 0, 0, -3, 0, 5, 9], [0, -3, 0, 0, -7, 0, 2, 11], [0, -4, 0, 0, 1, 0, -4, 7], [-9, 0, -6, 9, 0, -27, 0, 0], [14, 0, -23, -1, 0, 9, 0, 0], [0, -4, 0, 0, -4, 0, 0, 7], [0, 4, 0, 0, 7, 0, 4, 9], [0, -4, 0, 0, -11, 0, -5, 2], [-29, 0, 14, -8, 0, -15, 0, 0], [7, 0, 4, -38, 0, 8, 0, 0], [24, 0, 4, 0, 0, 18, 0, 0], [0, 9, 0, 0, 3, 0, 11, 10], [0, -17, 0, 0, 5, 0, -17, -5], [11, 0, -14, -11, 0, -16, 0, 0], [0, 3, 0, 0, 0, 0, 7, 5], [0, 17, 0, 0, 3, 0, 12, -5], [-4, 0, -20, -19, 0, 24, 0, 0], [33, 0, -3, 4, 0, 27, 0, 0], [0, -13, 0, 0, -8, 0, -16, -1], [-10, 0, -15, -4, 0, 3, 0, 0], [0, 13, 0, 0, 6, 0, 7, -7], [0, -21, 0, 0, -9, 0, -8, 9], [-21, 0, 7, 35, 0, -24, 0, 0], [-18, 0, 8, 38, 0, -16, 0, 0], [0, -1, 0, 0, -8, 0, 7, 11], [0, 0, 0, 0, 5, 0, -15, 5], [0, 0, 0, 0, -1, 0, -1, 8], [-40, 0, -5, 11, 0, -26, 0, 0], [-3, 0, -8, -11, 0, 7, 0, 0], [0, -11, 0, 0, 10, 0, -5, 1], [-4, 0, -7, 22, 0, 4, 0, 0], [0, -5, 0, 0, -2, 0, 6, -18], [-29, 0, -19, 10, 0, -18, 0, 0], [0, 6, 0, 0, 3, 0, -16, 8], [0, -6, 0, 0, -3, 0, 9, -3], [-23, 0, 20, -12, 0, -37, 0, 0], [0, 2, 0, 0, 9, 0, 10, -9], [-2, 0, 31, 4, 0, -15, 0, 0], [-47, 0, 4, 5, 0, -33, 0, 0], [30, 0, -5, 1, 0, 4, 0, 0], [0, -19, 0, 0, 0, 0, -16, 1], [-26, 0, 5, -27, 0, 2, 0, 0], [0, 18, 0, 0, -12, 0, 7, 7], [15, 0, -3, -3, 0, -22, 0, 0], [0, 7, 0, 0, -8, 0, 6, 1], [0, 0, 0, 0, 13, 0, 26, 4], [0, 5, 0, 0, 13, 0, 13, 5], [0, -5, 0, 0, -4, 0, 6, 7], [-10, 0, -9, -3, 0, 2, 0, 0], [-23, 0, -10, -5, 0, -6, 0, 0], [0, 0, -15, 0, 0, -7, 0, 0], [0, -24, 0, 0, -11, 0, 4, -5], [0, 9, 0, 0, 10, 0, -7, 10], [-8, 0, -4, -10, 0, 16, 0, 0], [0, -23, 0, 0, 1, 0, 2, -9], [10, 0, -10, -17, 0, 22, 0, 0], [0, -6, 0, 0, 12, 0, -12, -6], [-7, 0, 8, 1, 0, 15, 0, 0], [0, -9, 0, 0, 13, 0, -1, -8], [0, 12, 0, 0, 0, 0, 1, 15], [-8, 0, 20, 7, 0, -12, 0, 0], [0, 0, 0, 0, -16, 0, 7, 5], [0, -6, 0, 0, -8, 0, -16, 7], [-12, 0, 2, -6, 0, -4, 0, 0], [-17, 0, -15, 24, 0, -12, 0, 0], [0, -15, 0, 0, -2, 0, 2, 1], [-17, 0, -9, 16, 0, 16, 0, 0], [-27, 0, -14, 6, 0, -29, 0, 0], [0, -18, 0, 0, -5, 0, -7, -1], [-21, 0, -9, 2, 0, 8, 0, 0], [-23, 0, -17, 9, 0, -28, 0, 0], [0, 16, 0, 0, -2, 0, 10, -4], [10, 0, -3, 16, 0, 10, 0, 0], [0, 1, 0, 0, 0, 0, 9, -4], [0, 3, 0, 0, 2, 0, -2, 0], [3, 0, -16, -14, 0, 32, 0, 0], [0, -11, 0, 0, 3, 0, 14, -4], [-1, 0, 8, 2, 0, 4, 0, 0], [10, 0, -7, -10, 0, 46, 0, 0], [22, 0, 13, -2, 0, -16, 0, 0], [35, 0, -5, -8, 0, 8, 0, 0], [-8, 0, -11, -20, 0, 12, 0, 0], [-49, 0, 2, 20, 0, -24, 0, 0], [0, -2, 0, 0, -14, 0, -18, 3], [10, 0, -2, 1, 0, 41, 0, 0], [24, 0, -13, -17, 0, 12, 0, 0], [0, 1, 0, 0, -1, 0, 15, 7], [0, 4, 0, 0, 15, 0, 3, -5], [13, 0, 23, 0, 0, -4, 0, 0], [0, -3, 0, 0, 2, 0, 0, 15], [-5, 0, -34, -8, 0, 42, 0, 0], [-48, 0, 0, 7, 0, -35, 0, 0], [0, -1, 0, 0, -1, 0, 12, 0], [12, 0, 14, 0, 0, -33, 0, 0], [0, -12, 0, 0, -12, 0, 8, -4], [-11, 0, 29, 0, 0, -16, 0, 0], [0, -13, 0, 0, -15, 0, -24, 1], [0, -7, 0, 0, -8, 0, -18, 2], [0, 8, 0, 0, 4, 0, -16, 13], [0, 12, 0, 0, 4, 0, -7, 18], [0, 15, 0, 0, 8, 0, 9, -2], [23, 0, 7, -9, 0, 13, 0, 0], [0, 0, 0, 0, 8, 0, -14, 9], [-4, 0, 16, -12, 0, -36, 0, 0], [0, -10, 0, 0, -8, 0, -2, 2], [0, 1, 0, 0, 7, 0, -12, -3], [-7, 0, 29, 1, 0, -8, 0, 0], [-30, 0, -31, -3, 0, 29, 0, 0], [0, 0, 0, 0, -11, 0, -21, -3], [-17, 0, -10, 18, 0, 12, 0, 0], [0, -1, 0, 0, 2, 0, -7, 14], [3, 0, 37, -20, 0, -4, 0, 0], [2, 0, 23, 17, 0, 6, 0, 0], [0, 8, 0, 0, -5, 0, 1, 5], [-11, 0, 17, 4, 0, 12, 0, 0], [0, -9, 0, 0, 9, 0, 9, 9], [-15, 0, 23, 3, 0, -20, 0, 0], [0, -4, 0, 0, 0, 0, -27, 5], [0, -6, 0, 0, -3, 0, 1, -2], [0, 14, 0, 0, -4, 0, 11, 6], [-2, 0, -13, -14, 0, 48, 0, 0], [0, -11, 0, 0, 2, 0, -7, -1], [5, 0, 3, 29, 0, -10, 0, 0], [0, -12, 0, 0, -13, 0, -22, 5], [0, -10, 0, 0, 9, 0, -4, -14], [14, 0, -18, -7, 0, -19, 0, 0], [0, 0, 0, 0, 1, 0, -1, 4], [-42, 0, 14, 19, 0, -33, 0, 0], [1, 0, 1, 19, 0, 11, 0, 0], [0, -3, 0, 0, -9, 0, -4, -2], [-17, 0, 18, 22, 0, -17, 0, 0], [0, -27, 0, 0, -5, 0, -9, 10], [62, 0, -8, -7, 0, 3, 0, 0], [38, 0, 4, -4, 0, -20, 0, 0], [-9, 0, -42, -12, 0, 57, 0, 0], [0, 11, 0, 0, 14, 0, 14, 3], [0, 11, 0, 0, 9, 0, -6, 0], [-4, 0, 20, -6, 0, 8, 0, 0], [0, 26, 0, 0, 8, 0, 11, -20], [14, 0, -38, -8, 0, 5, 0, 0], [0, 6, 0, 0, 6, 0, 27, -3], [-4, 0, 3, 10, 0, 19, 0, 0], [0, 18, 0, 0, -4, 0, 15, 6], [8, 0, -3, 6, 0, 0, 0, 0], [0, 7, 0, 0, -6, 0, 22, -4], [23, 0, -31, 14, 0, 35, 0, 0], [-42, 0, -6, -21, 0, 1, 0, 0], [-6, 0, 4, 14, 0, -42, 0, 0], [0, -1, 0, 0, 10, 0, 10, 10], [0, -12, 0, 0, -13, 0, -20, 1], [2, 0, -8, -21, 0, 1, 0, 0], [0, -4, 0, 0, 8, 0, -9, -4], [0, -3, 0, 0, 10, 0, 8, -8], [-14, 0, 43, 1, 0, -23, 0, 0], [-15, 0, 5, 11, 0, -47, 0, 0], [0, -2, 0, 0, -18, 0, 2, 2], [0, 11, 0, 0, 9, 0, -2, -11], [12, 0, 14, -13, 0, 17, 0, 0], [59, 0, 19, -35, 0, 41, 0, 0], [0, 4, 0, 0, -5, 0, 4, -7], [-24, 0, -19, -17, 0, 33, 0, 0], [5, 0, -7, 24, 0, 20, 0, 0], [0, -19, 0, 0, -7, 0, -9, -4], [0, -10, 0, 0, -5, 0, -18, -4], [0, -13, 0, 0, 2, 0, 4, -21], [0, -20, 0, 0, 4, 0, 2, 3], [0, 5, 0, 0, -20, 0, -12, 8], [40, 0, -10, -3, 0, -24, 0, 0], [0, 17, 0, 0, -11, 0, 2, 17], [0, 23, 0, 0, -1, 0, 14, 18], [0, -7, 0, 0, 1, 0, -8, -20], [-11, 0, 15, -21, 0, -15, 0, 0], [24, 0, 35, -19, 0, 1, 0, 0], [0, -4, 0, 0, -11, 0, -28, -7], [35, 0, -20, -45, 0, 25, 0, 0], [-10, 0, 46, 3, 0, -45, 0, 0], [0, 15, 0, 0, -3, 0, 17, -5], [22, 0, 42, -14, 0, -2, 0, 0], [0, -10, 0, 0, -5, 0, -28, -3], [0, 20, 0, 0, -17, 0, -1, 8], [32, 0, -8, 25, 0, 17, 0, 0], [8, 0, 5, -14, 0, 49, 0, 0], [0, -1, 0, 0, 22, 0, 4, -13], [0, 4, 0, 0, -12, 0, -5, -13], [12, 0, 5, -10, 0, -16, 0, 0], [0, 24, 0, 0, 8, 0, -3, 3], [-1, 0, -37, -10, 0, 34, 0, 0], [0, 7, 0, 0, 5, 0, 25, -12], [0, 7, 0, 0, -3, 0, 18, -16], [2, 0, -9, 22, 0, -8, 0, 0], [0, -4, 0, 0, 7, 0, -20, -1], [-51, 0, -11, 15, 0, -41, 0, 0], [0, -12, 0, 0, 7, 0, -6, 7], [17, 0, -19, 11, 0, 35, 0, 0], [-18, 0, 14, -18, 0, -7, 0, 0], [32, 0, -4, -9, 0, -11, 0, 0], [0, 11, 0, 0, 12, 0, 7, -11], [-24, 0, 14, 6, 0, -19, 0, 0], [0, -19, 0, 0, 0, 0, -20, 8], [0, -23, 0, 0, -10, 0, -13, 4], [-20, 0, 0, 21, 0, -47, 0, 0], [0, 25, 0, 0, 8, 0, 18, 10], [0, 0, -35, -8, 0, 26, 0, 0], [-34, 0, 3, -2, 0, -65, 0, 0], [0, -6, 0, 0, -8, 0, -33, 1], [20, 0, 18, -10, 0, 17, 0, 0], [0, -11, 0, 0, -15, 0, -11, -7], [23, 0, 13, -16, 0, 6, 0, 0], [0, 25, 0, 0, 22, 0, 3, 7], [14, 0, 16, -13, 0, -34, 0, 0], [0, -6, 0, 0, 4, 0, -3, 7], [0, -24, 0, 0, -7, 0, -15, 20], [-14, 0, 12, 8, 0, -34, 0, 0], [39, 0, 4, 12, 0, -23, 0, 0], [0, 7, 0, 0, -14, 0, -7, 9], [2, 0, -15, 16, 0, 37, 0, 0], [0, 14, 0, 0, 9, 0, 2, 10], [50, 0, -17, 28, 0, 30, 0, 0], [0, -3, 0, 0, 4, 0, -13, -7], [0, 18, 0, 0, -3, 0, 14, 2], [-5, 0, 8, 15, 0, 32, 0, 0], [-58, 0, -23, 21, 0, -39, 0, 0], [0, 24, 0, 0, 0, 0, 4, -7], [0, 1, 0, 0, 6, 0, -12, 5], [9, 0, -17, -23, 0, 50, 0, 0], [0, 16, 0, 0, -2, 0, 24, 16], [0, -15, 0, 0, -2, 0, -13, 13], [0, 30, 0, 0, 3, 0, 3, 0], [11, 0, -1, -34, 0, 33, 0, 0], [0, 15, 0, 0, -6, 0, -7, -7], [0, 27, 0, 0, 7, 0, 8, 6], [9, 0, 17, 3, 0, -4, 0, 0], [0, 16, 0, 0, 13, 0, 31, -5], [0, 2, 0, 0, -17, 0, -2, 11], [0, -9, 0, 0, -4, 0, -11, -2], [-13, 0, -21, -13, 0, -3, 0, 0], [0, -14, 0, 0, 9, 0, -13, -17]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_5625_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_5625_2_a_y();". // 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_5625_2_a_y(:prec:=8) chi := MakeCharacter_5625_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_5625_2_a_y();". // 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_5625_2_a_y( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_5625_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![145, 0, -180, 0, 80, 0, -15, 0, 1]>,<7,R![-5, -10, 0, 5, 1]>],Snew); return Vf; end function;