// Make newform 7200.2.d.r in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_7200_d();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_7200_d_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_7200_2_d_r();". // 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_7200_2_d_r();". // 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 := [8, -8, 6, -6, 3, -2, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0], [-2, 2, -1, 2, -1, 0], [8, -2, 4, -3, 0, -1], [-8, 10, -6, 3, -2, 1], [3, 0, 5, -1, 1, -1], [-12, 2, -6, 5, -2, 3]]; Rf_basisdens := [1, 1, 2, 2, 1, 2]; 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_7200_d();" function MakeCharacter_7200_d() N := 7200; order := 2; char_gens := [6751, 901, 6401, 577]; v := [2, 1, 2, 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; // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_7200_d_Hecke();" function MakeCharacter_7200_d_Hecke(Kf) N := 7200; order := 2; char_gens := [6751, 901, 6401, 577]; char_values := [[1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [-1, 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, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1], [0, 0, -1, 1, 0, 1], [0, 1, 0, 0, 0, 0], [0, 0, -1, -1, 0, 0], [0, 0, 1, 0, 0, -1], [0, 0, 1, 0, 0, -1], [0, 0, -1, 0, 0, 0], [2, 1, 0, 0, 0, 0], [1, 0, 0, 0, -1, 0], [3, 1, 0, 0, -1, 0], [1, 1, 0, 0, -1, 0], [0, 0, 1, -2, 0, -1], [-2, 0, 0, 0, 0, 0], [0, 0, 1, 1, 0, 1], [0, 0, -2, 2, 0, 2], [-4, 0, 0, 0, 0, 0], [-1, 1, 0, 0, 1, 0], [0, 0, 3, 0, 0, 0], [6, -1, 0, 0, 0, 0], [5, 1, 0, 0, -1, 0], [-5, 1, 0, 0, -1, 0], [0, 0, 3, 2, 0, 0], [0, 0, -1, 0, 0, 0], [0, 0, 0, 2, 0, -1], [7, -3, 0, 0, 1, 0], [0, 0, 0, 0, 0, -2], [0, 0, 3, 1, 0, 4], [0, 0, 4, 2, 0, -1], [0, 0, 1, -1, 0, -3], [0, 0, 5, -1, 0, 0], [0, 0, 5, 2, 0, -1], [0, 0, -1, 0, 0, 0], [-5, 2, 0, 0, -1, 0], [-9, -2, 0, 0, 1, 0], [7, -1, 0, 0, 1, 0], [0, 0, 3, -2, 0, -1], [-4, 0, 0, 0, 2, 0], [0, 0, -3, -1, 0, 1], [0, 0, 8, 0, 0, -2], [-8, 0, 0, 0, 0, 0], [0, 0, -5, 2, 0, 4], [-14, -2, 0, 0, 0, 0], [-13, 2, 0, 0, -1, 0], [0, 0, 5, 2, 0, 3], [0, 0, -6, -2, 0, -1], [0, 2, 0, 0, 0, 0], [0, 0, 4, 2, 0, -2], [0, 0, -7, 1, 0, 0], [-11, 1, 0, 0, -1, 0], [4, 0, 0, 0, -2, 0], [0, 0, -1, 3, 0, -3], [0, 0, -3, 3, 0, 0], [0, 0, 7, -2, 0, -1], [0, 0, 3, 0, 0, 4], [-21, -2, 0, 0, -1, 0], [3, -4, 0, 0, 1, 0], [4, 0, 0, 0, -2, 0], [-20, 0, 0, 0, 0, 0], [-2, -4, 0, 0, 0, 0], [-15, -3, 0, 0, -1, 0], [7, -1, 0, 0, -3, 0], [0, 0, -3, 2, 0, 0], [10, 2, 0, 0, -4, 0], [0, 0, -7, 4, 0, 3], [0, 0, 3, -4, 0, -4], [-6, 0, 0, 0, -2, 0], [0, 0, -6, 2, 0, 6], [0, 0, -1, -1, 0, 4], [1, -1, 0, 0, -1, 0], [0, 0, 2, -4, 0, -5], [9, 0, 0, 0, -1, 0], [0, 0, 1, -2, 0, 3], [0, 0, -1, 0, 0, -1], [0, 0, -5, 4, 0, 0], [-2, 1, 0, 0, -2, 0], [5, 1, 0, 0, 1, 0], [6, 4, 0, 0, 0, 0], [0, 0, -1, 3, 0, 1], [0, 0, 12, 2, 0, -2], [-1, 1, 0, 0, 1, 0], [0, 0, 1, -4, 0, 0], [1, 0, 0, 0, 1, 0], [-6, 4, 0, 0, -2, 0], [-18, -4, 0, 0, 0, 0], [0, 0, 5, -2, 0, 4], [0, 0, 7, 0, 0, -4], [0, 0, 10, 0, 0, -1], [-19, -3, 0, 0, -1, 0], [15, -3, 0, 0, 1, 0], [0, 0, 4, 2, 0, 3], [0, 0, -7, 1, 0, 1], [0, 0, -3, -4, 0, -1], [0, 0, 5, 2, 0, 3], [0, 0, 7, 4, 0, -8], [-10, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 0], [0, 0, -4, 4, 0, -2], [5, 1, 0, 0, 3, 0], [16, 0, 0, 0, 2, 0], [-10, -4, 0, 0, 2, 0], [15, -1, 0, 0, -1, 0], [0, 0, -7, 0, 0, -1], [0, 0, 3, -2, 0, -4], [9, 3, 0, 0, -1, 0], [0, 0, -1, 1, 0, 0], [-15, -5, 0, 0, -1, 0], [-2, 0, 0, 0, 4, 0], [0, 0, 8, -4, 0, -1], [4, -1, 0, 0, 0, 0], [0, 0, -9, 3, 0, 0], [0, 0, 9, 2, 0, -1], [16, 1, 0, 0, -2, 0], [8, 4, 0, 0, 2, 0], [1, 5, 0, 0, -1, 0], [0, 0, -3, 6, 0, 11], [-6, 2, 0, 0, -4, 0], [0, 0, -3, -3, 0, -7], [0, 0, 6, 2, 0, 6], [0, 0, 9, 0, 0, 0], [-6, 2, 0, 0, 4, 0], [-27, 1, 0, 0, -1, 0], [0, 0, 5, 0, 0, -5], [0, 0, 9, -4, 0, -4], [0, 0, 8, 2, 0, 2], [-14, 0, 0, 0, 2, 0], [0, 0, 0, -6, 0, -9], [9, 2, 0, 0, -5, 0], [0, 0, -11, 8, 0, 7], [0, 0, 11, 4, 0, -1], [-14, 5, 0, 0, 0, 0], [21, 2, 0, 0, -1, 0], [-4, 6, 0, 0, 2, 0], [-2, -6, 0, 0, 0, 0], [-2, -2, 0, 0, 0, 0], [-10, -6, 0, 0, -2, 0], [10, 0, 0, 0, 0, 0], [16, -8, 0, 0, 2, 0], [0, 0, 5, -6, 0, 3], [0, 0, -13, -4, 0, 4], [0, 0, -2, 6, 0, 3], [20, 0, 0, 0, 0, 0], [0, 0, -16, 0, 0, -2], [11, -3, 0, 0, 5, 0], [-8, 3, 0, 0, 4, 0], [0, 0, -3, -3, 0, -4], [0, 0, -3, 2, 0, -5], [0, 0, -5, 6, 0, 7], [3, 2, 0, 0, -3, 0], [3, -7, 0, 0, -1, 0], [7, -9, 0, 0, 1, 0], [0, 0, -5, -6, 0, -5], [14, 6, 0, 0, -2, 0], [5, -7, 0, 0, -1, 0], [28, -1, 0, 0, -2, 0], [-12, 2, 0, 0, 2, 0], [0, 0, 7, 0, 0, -12], [0, 0, 7, 0, 0, 4], [4, -4, 0, 0, 0, 0], [0, 0, 11, -3, 0, 4], [0, 0, 6, -2, 0, 7], [0, 0, -9, -5, 0, 5], [0, 0, 3, -1, 0, 4], [0, 0, -1, 0, 0, 11], [11, -4, 0, 0, 3, 0], [4, 7, 0, 0, 0, 0], [22, -2, 0, 0, 4, 0], [14, -4, 0, 0, 4, 0], [0, 0, 5, -7, 0, 1], [0, 0, 6, -12, 0, -6], [12, -12, 0, 0, 4, 0], [0, 0, 5, 2, 0, 4], [28, -3, 0, 0, 2, 0], [-19, 3, 0, 0, -3, 0], [0, 0, -3, -2, 0, 7], [0, 0, -19, 0, 0, 4], [0, 0, 4, 0, 0, 3], [0, 0, 18, -4, 0, -2], [0, 0, -6, 2, 0, -5], [0, 0, -1, -3, 0, -7], [-12, 1, 0, 0, 4, 0], [0, 0, 3, -7, 0, 0], [0, 0, -21, 0, 0, -5], [0, 0, -13, -4, 0, 4], [-41, 4, 0, 0, -3, 0], [-3, 1, 0, 0, 3, 0], [0, -10, 0, 0, 2, 0], [-6, 2, 0, 0, -2, 0], [0, 0, 3, -6, 0, -8], [-18, -2, 0, 0, 6, 0], [0, 0, 5, 2, 0, 7], [0, 0, 19, 0, 0, 0], [-5, -3, 0, 0, 5, 0], [0, 0, 1, -5, 0, -4], [18, -4, 0, 0, 4, 0], [22, -9, 0, 0, 2, 0], [0, 0, -15, 3, 0, 8], [0, 0, 23, -4, 0, -1], [0, 0, -7, 8, 0, 12], [-16, -3, 0, 0, 6, 0], [-2, 3, 0, 0, 2, 0], [-8, 4, 0, 0, -6, 0], [0, 0, -13, -1, 0, 5], [18, 8, 0, 0, 4, 0], [-2, -7, 0, 0, -4, 0], [-2, -4, 0, 0, -6, 0], [5, 3, 0, 0, 5, 0], [0, 0, 1, 0, 0, -9], [0, 0, 5, -6, 0, 4], [0, 0, 5, 4, 0, -8], [0, 0, 28, 2, 0, -1], [-22, -2, 0, 0, 2, 0], [-18, 8, 0, 0, -2, 0], [4, -8, 0, 0, 6, 0], [0, 0, -2, 4, 0, 3], [3, -5, 0, 0, 3, 0], [0, 0, 7, 0, 0, -9], [18, -12, 0, 0, 4, 0], [0, 0, 0, -2, 0, 10], [-21, 2, 0, 0, -1, 0], [-5, 9, 0, 0, -1, 0], [0, 0, 2, -8, 0, -9], [15, 7, 0, 0, -3, 0], [0, 0, 10, -4, 0, -10], [0, 0, -7, -1, 0, 16], [32, -2, 0, 0, 4, 0], [0, 0, 6, 0, 0, -1], [0, 0, -3, -5, 0, -7], [-46, 3, 0, 0, -2, 0], [0, 0, 9, 2, 0, 7], [-13, -6, 0, 0, -1, 0], [-2, -2, 0, 0, -4, 0], [50, 6, 0, 0, 2, 0], [0, 0, -3, 4, 0, 7], [16, 8, 0, 0, 2, 0], [14, 6, 0, 0, -4, 0], [0, 0, 7, 3, 0, -11], [19, -11, 0, 0, 5, 0], [-12, 6, 0, 0, -4, 0], [0, 0, -11, -8, 0, -9], [0, 0, 14, -10, 0, -1], [0, 0, 2, 2, 0, 10], [0, 0, 19, -1, 0, -8], [-44, -6, 0, 0, 0, 0], [0, 0, -2, -4, 0, 3], [0, 0, -19, 1, 0, 1], [0, 0, 13, 4, 0, 3], [0, 0, -9, -6, 0, -5], [-28, 13, 0, 0, -4, 0], [6, 4, 0, 0, 0, 0], [0, 0, 1, -8, 0, 3], [20, 4, 0, 0, 6, 0], [-4, 0, 0, 0, -2, 0], [0, 0, -23, -3, 0, 1], [0, 0, 12, 0, 0, 10], [-7, 5, 0, 0, -1, 0], [-24, 4, 0, 0, -6, 0], [0, 0, -15, -4, 0, -12], [0, 0, 6, -10, 0, -17], [8, 4, 0, 0, -4, 0], [0, 0, 16, 2, 0, -2], [49, 4, 0, 0, -1, 0], [0, 0, -21, -3, 0, -4], [0, 0, -15, -4, 0, -1], [0, 0, 9, 4, 0, -4], [43, -1, 0, 0, 3, 0], [-2, -6, 0, 0, -2, 0], [-30, -4, 0, 0, 0, 0], [0, 0, 2, 8, 0, 2], [27, 3, 0, 0, 5, 0], [0, 0, 5, 2, 0, 4], [18, 3, 0, 0, -4, 0], [0, 0, 13, 2, 0, 8], [0, 0, 4, 6, 0, -9], [-16, -2, 0, 0, 0, 0], [0, 0, -6, -2, 0, -2], [4, 8, 0, 0, -2, 0], [0, 0, -15, -5, 0, 9], [0, 0, -3, -4, 0, -5], [17, 6, 0, 0, 5, 0], [0, 0, -11, 6, 0, -1], [0, 0, -18, 0, 0, 6], [-36, 8, 0, 0, 0, 0], [-22, -2, 0, 0, 6, 0], [0, 0, -5, 8, 0, -4], [-26, 2, 0, 0, 0, 0], [5, 2, 0, 0, 1, 0], [23, 3, 0, 0, -1, 0], [0, 0, -17, -4, 0, 8], [5, 7, 0, 0, -5, 0], [0, 0, -3, 13, 0, 0], [0, 0, 9, 11, 0, 1], [-17, -6, 0, 0, 9, 0], [0, 0, 5, 12, 0, 0], [-8, -3, 0, 0, -2, 0], [-40, 0, 0, 0, 6, 0], [0, 0, 31, 3, 0, 1], [49, -3, 0, 0, -1, 0], [0, 0, 5, 10, 0, 8], [10, 12, 0, 0, 0, 0], [19, -2, 0, 0, -1, 0], [-41, -3, 0, 0, 1, 0], [0, 0, 1, 4, 0, 15], [0, 0, -11, -4, 0, 12], [-17, -9, 0, 0, -3, 0], [0, 0, 12, 2, 0, 10], [22, 4, 0, 0, -10, 0], [-43, -4, 0, 0, 3, 0], [0, 0, -25, 6, 0, 3], [0, 0, 9, 0, 0, 0], [-11, -7, 0, 0, -7, 0], [-6, 6, 0, 0, 2, 0], [0, 0, 19, -4, 0, -5], [46, -2, 0, 0, 4, 0], [0, 0, 9, -13, 0, -7], [-20, 14, 0, 0, -8, 0], [0, 0, -19, 0, 0, 4], [3, 7, 0, 0, -5, 0], [0, 0, -3, 0, 0, 11], [0, 0, 3, -10, 0, -8], [0, 0, 5, 8, 0, 12], [0, 0, -4, 2, 0, 15], [0, 0, -9, 7, 0, 16], [-6, 8, 0, 0, 4, 0], [0, 0, 13, 6, 0, -9], [-9, -5, 0, 0, 1, 0], [0, 0, -21, 4, 0, -1], [-10, -4, 0, 0, -4, 0], [0, 0, -16, -6, 0, -2], [-24, 4, 0, 0, 2, 0], [40, 7, 0, 0, 2, 0], [-21, -3, 0, 0, 5, 0], [0, 0, 5, 6, 0, 19], [-3, 9, 0, 0, -1, 0], [0, 0, -6, -4, 0, -18], [0, 0, 11, 7, 0, -12], [-6, 0, 0, 0, 0, 0], [0, 0, 24, -2, 0, 3], [-11, 4, 0, 0, -5, 0], [0, 0, -7, 9, 0, 12], [0, 0, 9, 4, 0, -12], [-5, -8, 0, 0, 3, 0], [0, 8, 0, 0, 2, 0], [0, 0, -1, -3, 0, -19], [0, 0, 8, -14, 0, -18], [8, -4, 0, 0, 4, 0], [-56, -2, 0, 0, -4, 0], [38, -4, 0, 0, -4, 0], [0, 0, -3, -10, 0, -13], [0, 0, -11, -10, 0, -4], [0, 0, -7, 12, 0, 4], [0, 0, -26, 10, 0, 7], [0, 0, -26, 6, 0, 6], [0, 0, -11, 5, 0, 0], [-42, -6, 0, 0, 2, 0], [0, 0, -15, -5, 0, -19], [0, 0, 31, 1, 0, 8], [0, 0, 31, 2, 0, -5], [14, -11, 0, 0, 6, 0], [-3, 9, 0, 0, 1, 0], [0, 0, -5, -8, 0, -5], [0, 0, 31, -1, 0, 1], [20, -4, 0, 0, 0, 0], [0, 0, 3, -14, 0, 0], [-18, 0, 0, 0, -8, 0], [0, 0, -6, 16, 0, 15], [24, 6, 0, 0, -6, 0], [0, 0, -9, -5, 0, -11], [0, 0, 29, -2, 0, -9], [0, 0, 15, 4, 0, -5], [0, 0, 5, 4, 0, 20], [55, -2, 0, 0, -5, 0], [46, -1, 0, 0, 2, 0], [0, 0, 11, 9, 0, 9], [-51, -3, 0, 0, -1, 0], [0, 0, -9, 12, 0, 8], [40, 2, 0, 0, 6, 0], [0, 0, -21, 4, 0, 8], [0, 0, -13, -4, 0, 12], [0, 0, 27, -7, 0, -4], [0, 0, 0, -8, 0, -1], [0, 0, -15, -3, 0, 0], [0, 0, 1, -10, 0, 7], [0, 0, 7, -10, 0, -5], [34, -3, 0, 0, 0, 0], [-21, 8, 0, 0, -7, 0], [-9, -1, 0, 0, 9, 0], [0, 0, 9, 12, 0, -9], [14, 6, 0, 0, -8, 0], [32, 12, 0, 0, 2, 0], [0, 0, 1, -3, 0, -19], [16, 4, 0, 0, -4, 0], [52, -4, 0, 0, -4, 0], [0, 0, 11, -6, 0, -12], [-17, -4, 0, 0, 7, 0], [8, 8, 0, 0, 2, 0], [0, 0, 29, 4, 0, -1], [0, 0, 9, 0, 0, 16], [0, 0, -2, -4, 0, -2], [0, 0, -23, -5, 0, -8], [0, 0, 24, 2, 0, 3], [0, 0, -19, -9, 0, -8], [0, 0, -7, -12, 0, -4], [-22, -5, 0, 0, -4, 0], [28, 9, 0, 0, 4, 0], [7, -5, 0, 0, 7, 0], [13, 5, 0, 0, -5, 0], [0, 0, 19, -9, 0, -3], [0, 0, 29, -6, 0, -4], [46, -4, 0, 0, 0, 0], [29, 13, 0, 0, 7, 0], [0, 0, -11, 16, 0, 3], [0, 0, 21, 12, 0, -8], [0, 0, -19, 0, 0, -8], [-32, 16, 0, 0, -8, 0], [0, 0, 12, 8, 0, -1], [0, 0, 11, 9, 0, -8], [0, 0, 25, 4, 0, 7], [0, 0, -7, -12, 0, -8], [8, -5, 0, 0, 4, 0], [0, 0, -25, -4, 0, -9], [0, 0, -5, 13, 0, 21], [0, 0, 21, -6, 0, -8], [-34, 12, 0, 0, -8, 0], [-41, 15, 0, 0, -3, 0], [-12, 14, 0, 0, -6, 0], [0, 0, 9, 4, 0, -9], [73, 3, 0, 0, -1, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_7200_d_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_7200_2_d_r();". // 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_7200_2_d_r(:prec:=6) chi := MakeCharacter_7200_d(); 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_7200_2_d_r();". // 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_7200_2_d_r( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7200_d(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![256, 0, 320, 0, 36, 0, 1]>,<11,R![4096, 0, 1088, 0, 64, 0, 1]>,<13,R![16, -28, 0, 1]>,<37,R![256, -60, -4, 1]>,<41,R![232, -36, -10, 1]>,<53,R![2, 1]>],Snew); return Vf; end function;