// Make newform 825.2.bx.f in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_825_bx();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_825_bx_Hecke();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_825_2_bx_f();". // 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_825_2_bx_f();". // 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, 0, -1, 0, 15, 0, -59, 0, 104, 0, -59, 0, 15, 0, -1, 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, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-63, 0, -622, 0, 1045, 0, 772, 0, -908, 0, 289, 0, 2, 0, 21, 0], [-119, 0, 1498, 0, -1995, 0, 2436, 0, -1260, 0, 329, 0, -22, 0, 21, 0], [0, 119, 0, -1498, 0, 1995, 0, -2436, 0, 1260, 0, -329, 0, 22, 0, -21], [-40, 0, 137, 0, -671, 0, 804, 0, -384, 0, 76, 0, -11, 0, 5, 0], [91, 0, -246, 0, 475, 0, -1604, 0, 1084, 0, -309, 0, 10, 0, -21, 0], [30, 0, 155, 0, -1107, 0, 1332, 0, -648, 0, 138, 0, -17, 0, 9, 0], [0, 30, 0, 155, 0, -1107, 0, 1332, 0, -648, 0, 138, 0, -17, 0, 9], [0, -16, 0, -685, 0, 1867, 0, -1748, 0, 736, 0, -148, 0, 23, 0, -9], [77, 0, 466, 0, -1511, 0, 4564, 0, -2972, 0, 845, 0, -30, 0, 57, 0], [0, 685, 0, -1990, 0, 2945, 0, -5644, 0, 3428, 0, -947, 0, 42, 0, -63], [-58, 0, 201, 0, -101, 0, -1108, 0, 824, 0, -246, 0, 5, 0, -17, 0], [0, -58, 0, 201, 0, -101, 0, -1108, 0, 824, 0, -246, 0, 5, 0, -17], [0, -543, 0, 2666, 0, -9107, 0, 10980, 0, -5388, 0, 1185, 0, -134, 0, 77], [0, 97, 0, -71, 0, -1556, 0, 3776, 0, -2284, 0, 589, 0, -35, 0, 40]]; Rf_basisdens := [1, 1, 384, 384, 384, 96, 192, 96, 96, 96, 384, 384, 96, 96, 384, 96]; 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_825_bx();" function MakeCharacter_825_bx() N := 825; order := 10; char_gens := [551, 727, 376]; v := [10, 5, 4]; // 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; // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_825_bx_Hecke();" function MakeCharacter_825_bx_Hecke(Kf) N := 825; order := 10; char_gens := [551, 727, 376]; char_values := [[1, 0, 0, 0, 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], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0]]; assert UnitGenerators(DirichletGroup(N)) eq char_gens; values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field F := Universe(values);// the Hecke field chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values); return chi; 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0], [0, 0, 0, 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, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, -1, 0, 0], [1, 0, -2, -1, 0, -3, -3, -2, 0, 0, -2, 0, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 1, -1], [0, 2, 0, 0, 2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 1, 0, 2, 0, 0, 0], [0, 0, 0, 0, -4, 0, 0, 0, 1, -1, 0, -2, 0, -4, -2, 1], [0, 0, 0, -3, 0, 1, 1, -6, 0, 0, -5, 0, 3, 0, 0, 0], [5, 0, 1, -1, 0, -4, 1, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 3, 0, 0, 0, -3, 5, 0, 1, 0, -3, 0, -1], [-4, 0, 1, 4, 0, 0, 0, 0, 0, 0, 1, 0, -8, 0, 0, 0], [0, 0, 0, 0, 5, 0, 0, 0, 0, 1, 0, 4, 0, 5, 4, -1], [0, -5, 0, 0, 0, 0, 0, 0, 1, -2, 0, 0, 0, -1, -2, 0], [0, 3, 0, 0, 3, 0, 0, 0, 0, 0, 0, 1, 0, 3, -6, 6], [0, 0, 0, -2, 0, 2, 2, -6, 0, 0, -3, 0, 2, 0, 0, 0], [3, 0, -6, 2, 0, 1, 0, 2, 0, 0, 1, 0, 3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, -4, 0, 3, 0, 0, 3, 4], [4, 0, -2, -9, 0, -5, 0, -9, 0, 0, -5, 0, 4, 0, 0, 0], [0, 5, 0, 0, 5, 0, 0, 0, -5, 3, 0, 4, 0, 5, 0, -4], [5, 0, 4, -4, 0, 2, 4, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 1, 0, 0, 0, -3, 5, 0, 5, 0, 0, 0, 0], [5, 0, -8, 0, 0, -8, -4, -3, 0, 0, -4, 0, 3, 0, 0, 0], [0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 7, 0, -5, 2, -2], [-12, 0, -2, 12, 0, -3, -2, 12, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 9, 0, 0, 0, -9, 6, 0, 2, 0, 4, 0, -2], [0, 8, 0, 0, 0, 0, 0, 0, -3, -4, 0, 0, 0, 3, -4, -2], [1, 0, 1, 0, 0, 1, 0, -7, 0, 0, 0, 0, 7, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, -13, 0, 0, 0, -5, -13, 5], [0, -1, 0, 0, -2, 0, 0, 0, 1, -1, 0, -1, 0, 0, -7, 0], [-9, 0, 6, 0, 0, 6, 5, 4, 0, 0, 5, 0, -4, 0, 0, 0], [0, 0, 0, 0, -11, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 0], [0, 0, 0, 3, 0, 13, 13, 11, 0, 0, 2, 0, -3, 0, 0, 0], [-1, 0, -7, 7, 0, -4, 0, 7, 0, 0, -4, 0, -1, 0, 0, 0], [13, 0, 4, -13, 0, 0, 8, 0, 0, 0, 4, 0, 6, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, 0, -3, -9, 0, 0, 0, 3, -9, 4], [0, -6, 0, 0, -6, 0, 0, 0, 0, 0, 0, 5, 0, -6, 9, -9], [0, 4, 0, 0, 4, 0, 0, 0, 0, 0, 0, 6, 0, 10, 12, -12], [0, 1, 0, 0, 0, 0, 0, 0, -4, 14, 0, 0, 0, 4, 14, -7], [4, 0, -5, -4, 0, 0, 1, 0, 0, 0, -5, 0, 6, 0, 0, 0], [9, 0, 6, -15, 0, 5, 0, -15, 0, 0, 5, 0, 9, 0, 0, 0], [0, 0, 0, 7, 0, 3, 3, 8, 0, 0, 7, 0, -7, 0, 0, 0], [0, 8, 0, 0, 12, 0, 0, 0, -8, 12, 0, 12, 0, 0, 3, 0], [0, 0, 0, 0, -9, 0, 0, 0, 0, -3, 0, 0, 0, -9, 0, 3], [-7, 0, 8, 0, 0, 8, 13, 12, 0, 0, 13, 0, -12, 0, 0, 0], [-2, 0, 9, -4, 0, 9, 9, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 11, -7], [0, 7, 0, 0, 1, 0, 0, 0, -1, 4, 0, -2, 0, 7, 0, 2], [5, 0, 3, -12, 0, 3, 3, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, -2, 0, 0, 0, 0, 0, 0, -2, 0, 9, 11, -11], [-4, 0, 11, 4, 0, 0, -2, 0, 0, 0, 11, 0, 0, 0, 0, 0], [-13, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, -6, 0, 0, 0], [0, 0, 2, 9, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, -3, 0, 0, 0, 3, 10, 0, 6, 0, -4, 0, -6], [0, 0, 0, 0, -7, 0, 0, 0, -4, -9, 0, -16, 0, -7, -16, 9], [-5, 0, 11, 6, 0, 5, 0, 6, 0, 0, 5, 0, -5, 0, 0, 0], [0, 0, 0, -18, 0, 4, 4, -18, 0, 0, -5, 0, 18, 0, 0, 0], [0, -7, 0, 0, -7, 0, 0, 0, 0, 0, 0, -16, 0, -8, -1, 1], [13, 0, -14, -9, 0, -2, 0, -9, 0, 0, -2, 0, 13, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, -6, -2, 0, 0, 0, 6, -2, -2], [0, -10, 0, 0, 3, 0, 0, 0, -3, 11, 0, 6, 0, -10, 0, -6], [0, 0, 0, 0, 7, 0, 0, 0, 6, 19, 0, 17, 0, 7, 17, -19], [-10, 0, 1, 10, 0, 0, 6, 0, 0, 0, 1, 0, -7, 0, 0, 0], [0, 2, 0, 0, -6, 0, 0, 0, -2, -1, 0, -1, 0, 0, -9, 0], [0, -6, 0, 0, -6, 0, 0, 0, 0, 0, 0, -10, 0, -2, 0, 0], [0, 0, 5, 0, 0, 5, 0, 8, 0, 0, 0, 0, -8, 0, 0, 0], [0, 6, 0, 0, 9, 0, 0, 0, -9, -4, 0, -8, 0, 6, 0, 8], [0, 6, 0, 0, 7, 0, 0, 0, -6, -17, 0, -17, 0, 0, -22, 0], [-3, 0, -3, 3, 0, 0, -3, 0, 0, 0, -3, 0, 6, 0, 0, 0], [0, 0, 0, 0, 10, 0, 0, 0, 10, -4, 0, 5, 0, 10, 5, 4], [0, 0, 0, 2, 0, -19, -19, -10, 0, 0, 0, 0, -2, 0, 0, 0], [0, 5, 0, 0, 12, 0, 0, 0, -12, 16, 0, 4, 0, 5, 0, -4], [0, 0, 0, 0, -9, 0, 0, 0, 5, -15, 0, -12, 0, -9, -12, 15], [6, 0, 5, -19, 0, 5, 0, -19, 0, 0, 5, 0, 6, 0, 0, 0], [0, 15, 0, 0, 15, 0, 0, 0, 0, 0, 0, 15, 0, 9, 2, -2], [0, 0, 0, 2, 0, 5, 5, 7, 0, 0, 14, 0, -2, 0, 0, 0], [0, 0, 0, 0, 4, 0, 0, 0, -5, 20, 0, 8, 0, 4, 8, -20], [5, 0, -7, -6, 0, -4, 0, -6, 0, 0, -4, 0, 5, 0, 0, 0], [0, 0, 8, -2, 0, 4, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-22, 0, -5, 0, 0, -5, -12, 16, 0, 0, -12, 0, -16, 0, 0, 0], [-8, 0, -9, 8, 0, 0, -10, 0, 0, 0, -9, 0, -3, 0, 0, 0], [-13, 0, 4, 4, 0, -4, 4, 13, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 11, 0, 0, 0, -11, 1, 0, -9, 0, 4, 0, 9], [-19, 0, -5, 0, 0, -5, -1, 13, 0, 0, -1, 0, -13, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -10, -1, 0, 0, 0, 10, -1, -4], [6, 0, -21, 9, 0, -14, -21, -6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 15, 0, 0, 3, 0, 0, 0, -15, 7, 0, 7, 0, 0, -12, 0], [-12, 0, -20, 0, 0, -20, -22, -5, 0, 0, -22, 0, 5, 0, 0, 0], [0, 0, 0, 0, -16, 0, 0, 0, 18, -26, 0, -23, 0, -16, -23, 26], [0, -4, 0, 0, 19, 0, 0, 0, 4, 16, 0, 16, 0, 0, -5, 0], [-6, 0, 2, -6, 0, -3, 0, -6, 0, 0, -3, 0, -6, 0, 0, 0], [0, -10, 0, 0, 0, 0, 0, 0, 10, -9, 0, 0, 0, -10, -9, -6], [0, 0, 0, -7, 0, 3, 3, 11, 0, 0, 0, 0, 7, 0, 0, 0], [0, 0, 0, 21, 0, 1, 1, 4, 0, 0, -1, 0, -21, 0, 0, 0], [0, 20, 0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 8, 0, -11], [9, 0, -2, -9, 0, 0, 25, 0, 0, 0, -2, 0, -7, 0, 0, 0], [0, 0, 0, 10, 0, 2, 2, -9, 0, 0, 3, 0, -10, 0, 0, 0], [0, -7, 0, 0, -5, 0, 0, 0, 7, -3, 0, -3, 0, 0, -5, 0], [-17, 0, -6, 9, 0, -3, -6, 17, 0, 0, 0, 0, 0, 0, 0, 0], [0, 14, 0, 0, 0, 0, 0, 0, -4, -9, 0, 0, 0, 4, -9, 23], [0, 3, 0, 0, -8, 0, 0, 0, 8, -22, 0, -14, 0, 3, 0, 14], [0, -5, 0, 0, -5, 0, 0, 0, 0, 0, 0, 5, 0, -3, -12, 12], [-6, 0, -1, 6, 0, 0, -1, 0, 0, 0, -1, 0, 10, 0, 0, 0], [18, 0, 14, 0, 0, 14, 13, -4, 0, 0, 13, 0, 4, 0, 0, 0], [0, 1, 0, 0, -17, 0, 0, 0, -1, -12, 0, -12, 0, 0, 6, 0], [0, -13, 0, 0, 2, 0, 0, 0, -2, 30, 0, 12, 0, -13, 0, -12], [0, 0, 0, 0, 10, 0, 0, 0, -4, 31, 0, 20, 0, 10, 20, -31], [22, 0, 6, -27, 0, 6, 0, -27, 0, 0, 6, 0, 22, 0, 0, 0], [0, 0, 0, 8, 0, -12, -12, 1, 0, 0, 11, 0, -8, 0, 0, 0], [0, -10, 0, 0, -10, 0, 0, 0, 0, 0, 0, 0, 0, -19, 8, -8], [0, 9, 0, 0, 0, 0, 0, 0, 10, -5, 0, 0, 0, -10, -5, 17], [0, 0, 0, 0, 11, 0, 0, 0, 1, -11, 0, 6, 0, 11, 6, 11], [-4, 0, -6, 4, 0, 0, -8, 0, 0, 0, -6, 0, 27, 0, 0, 0], [0, 0, 0, -18, 0, 0, 0, -22, 0, 0, -25, 0, 18, 0, 0, 0], [-7, 0, -3, 7, 0, 0, -14, 0, 0, 0, -3, 0, 7, 0, 0, 0], [0, -10, 0, 0, 8, 0, 0, 0, 10, -17, 0, -17, 0, 0, 4, 0], [0, 21, 0, 0, 21, 0, 0, 0, 0, 0, 0, 13, 0, 16, 4, -4], [0, -4, 0, 0, -4, 0, 0, 0, 4, 1, 0, -7, 0, -4, 0, 7], [-3, 0, 8, 0, 0, 8, -4, -4, 0, 0, -4, 0, 4, 0, 0, 0], [-15, 0, 12, 0, 0, 12, 12, 34, 0, 0, 12, 0, -34, 0, 0, 0], [0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 3, 0, -26, -4, 4], [0, -7, 0, 0, -30, 0, 0, 0, 7, -27, 0, -27, 0, 0, -16, 0], [0, 0, 0, 0, 26, 0, 0, 0, -10, 27, 0, 22, 0, 26, 22, -27], [-15, 0, -4, -9, 0, 5, -4, 15, 0, 0, 0, 0, 0, 0, 0, 0], [-8, 0, -2, 8, 0, 0, -11, 0, 0, 0, -2, 0, -20, 0, 0, 0], [-9, 0, -26, 14, 0, -4, 0, 14, 0, 0, -4, 0, -9, 0, 0, 0], [0, 0, 0, 19, 0, 7, 7, 28, 0, 0, 16, 0, -19, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 10, -32, 0, -28, 0, 0, -28, 32], [0, 20, 0, 0, 1, 0, 0, 0, -1, 6, 0, 7, 0, 20, 0, -7], [-26, 0, 17, 17, 0, 19, 17, 26, 0, 0, 0, 0, 0, 0, 0, 0], [0, -7, 0, 0, -5, 0, 0, 0, 7, 8, 0, 8, 0, 0, 17, 0], [30, 0, 8, -30, 0, 0, 13, 0, 0, 0, 8, 0, 22, 0, 0, 0], [0, -6, 0, 0, -6, 0, 0, 0, 0, 0, 0, -12, 0, -16, -10, 10], [-11, 0, -5, 27, 0, 12, -5, 11, 0, 0, 0, 0, 0, 0, 0, 0], [9, 0, 12, 0, 0, 12, 10, 4, 0, 0, 10, 0, -4, 0, 0, 0], [0, 10, 0, 0, 0, 0, 0, 0, -7, -26, 0, 0, 0, 7, -26, 9], [0, -1, 0, 0, -3, 0, 0, 0, 1, -12, 0, -12, 0, 0, 13, 0], [0, -5, 0, 0, 3, 0, 0, 0, 5, 0, 0, 0, 0, 0, 3, 0], [-36, 0, -4, 18, 0, 3, 0, 18, 0, 0, 3, 0, -36, 0, 0, 0], [-18, 0, -6, 18, 0, 0, -3, 0, 0, 0, -6, 0, 9, 0, 0, 0], [0, 0, 0, -9, 0, 6, 6, -2, 0, 0, -5, 0, 9, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 35, 0, 5, 12, -12], [0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 41, 0, -4, 8, -8], [0, 0, 0, -3, 0, 7, 7, 3, 0, 0, 1, 0, 3, 0, 0, 0], [14, 0, 20, -14, 0, 0, 0, 0, 0, 0, 20, 0, -15, 0, 0, 0], [0, -8, 0, 0, -1, 0, 0, 0, 8, 10, 0, 10, 0, 0, -17, 0], [0, 0, 0, 0, -3, 0, 0, 0, -10, 0, 0, -14, 0, -3, -14, 0], [9, 0, -11, 0, 0, -11, -13, 4, 0, 0, -13, 0, -4, 0, 0, 0], [0, 2, 0, 0, 9, 0, 0, 0, -2, 17, 0, 17, 0, 0, -6, 0], [0, 22, 0, 0, 0, 0, 0, 0, -1, -9, 0, 0, 0, 1, -9, 16], [-6, 0, 0, 0, 0, 0, 15, 27, 0, 0, 15, 0, -27, 0, 0, 0], [0, -18, 0, 0, 0, 0, 0, 0, -4, 17, 0, 0, 0, 4, 17, -6], [0, 5, 0, 0, 3, 0, 0, 0, -3, 16, 0, 14, 0, 5, 0, -14], [0, -2, 0, 0, -8, 0, 0, 0, 2, 16, 0, 16, 0, 0, 19, 0], [3, 0, 4, 7, 0, -10, 4, -3, 0, 0, 0, 0, 0, 0, 0, 0], [-14, 0, 10, 2, 0, 9, 0, 2, 0, 0, 9, 0, -14, 0, 0, 0], [29, 0, 14, -9, 0, -5, 0, -9, 0, 0, -5, 0, 29, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 29, 4, -4], [13, 0, 6, -17, 0, -4, 0, -17, 0, 0, -4, 0, 13, 0, 0, 0], [0, 0, 0, 0, 11, 0, 0, 0, -8, -11, 0, -10, 0, 11, -10, 11], [0, 13, 0, 0, -1, 0, 0, 0, 1, -24, 0, -21, 0, 13, 0, 21], [0, 0, 0, 0, 10, 0, 0, 0, -7, -12, 0, 5, 0, 10, 5, 12], [-4, 0, 24, 4, 0, 0, 6, 0, 0, 0, 24, 0, -10, 0, 0, 0], [0, -19, 0, 0, -19, 0, 0, 0, 0, 0, 0, 5, 0, -27, -7, 7], [0, -17, 0, 0, -18, 0, 0, 0, 18, -12, 0, -6, 0, -17, 0, 6], [13, 0, -8, 0, 0, -8, 11, -12, 0, 0, 11, 0, 12, 0, 0, 0], [0, 10, 0, 0, 16, 0, 0, 0, -16, -2, 0, -12, 0, 10, 0, 12]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_825_bx_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_825_2_bx_f();". // 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_825_2_bx_f(:prec:=16) chi := MakeCharacter_825_bx(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); S := BaseChange(S, Kf); maxprec := NextPrime(997) - 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_825_2_bx_f();". // 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_825_2_bx_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_825_bx(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![1, 0, 7, 0, 195, 0, -403, 0, 354, 0, -137, 0, 25, 0, -2, 0, 1]>,<13,R![14641, 0, 10527, 0, 24275, 0, 2397, 0, 6574, 0, -3477, 0, 725, 0, -42, 0, 1]>],Snew); return Vf; end function;