// Make newform 7448.2.a.bn in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_7448_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_7448_2_a_bn();". // 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_7448_2_a_bn();". // 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, -9, 8, 15, -6, -3, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [-5, 3, 13, -5, -3, 1], [-4, 19, 10, -9, -2, 1], [11, -13, -17, 7, 3, -1], [5, -9, -14, 6, 3, -1]]; Rf_basisdens := [1, 1, 2, 2, 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_7448_a();" function MakeCharacter_7448_a() N := 7448; order := 1; char_gens := [1863, 3725, 3041, 3137]; 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_7448_a_Hecke(Kf) return MakeCharacter_7448_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, 0, 0, 0, 0, 0], [1, -1, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0], [1, -1, 1, 0, 0, 1], [2, -1, 2, 0, -1, 1], [0, 0, 0, -1, 1, -1], [-1, 0, 0, 0, 0, 0], [0, 0, 0, -2, -1, 0], [0, -3, -2, -2, 0, 0], [2, 0, 1, 1, 0, 2], [-2, 1, 0, 1, 0, 1], [-4, -2, 0, 0, 0, 3], [0, 0, 1, -1, 0, 1], [5, 0, 2, 1, -1, 0], [-3, -2, -3, 0, 1, 0], [4, -1, -1, -1, -1, 0], [3, 4, 2, 3, 1, 0], [2, 1, 3, 1, 2, -1], [3, -1, 0, -1, -1, -1], [-2, -2, -3, -1, -1, 2], [-6, 2, -1, -1, 0, 3], [0, -2, -3, -1, -1, 0], [-3, 4, -1, 1, 3, 1], [-3, 3, -1, 1, 0, -2], [4, -2, -3, 0, -2, -1], [2, -3, -4, 0, -1, 1], [-4, 1, -3, -4, 0, 0], [1, -1, 0, 3, 1, 1], [-4, -2, -2, -3, 4, -1], [-3, -2, 1, 1, -1, 5], [4, -4, -2, -5, -1, 1], [6, 0, 3, 3, -2, 3], [-2, 6, -1, 0, 0, -3], [0, 0, -2, 0, 1, -2], [4, -11, -1, -5, -2, 1], [5, 4, 3, 4, 2, 0], [-1, 5, 3, 2, 1, -1], [-2, 5, 2, 2, 5, -1], [10, -2, -2, -4, 0, -2], [2, -3, -5, -3, 2, -3], [-4, 0, -1, 1, -6, 2], [2, 1, 4, -3, -4, 0], [4, -3, 3, 3, -4, 1], [-2, 1, 0, 3, 4, -4], [2, 4, -6, 1, 4, -2], [-4, 8, 0, 3, -2, -1], [2, -11, -4, -4, -1, 3], [0, 2, -3, -3, 4, -2], [14, 0, 3, 1, 0, 1], [-3, 1, 3, 0, 2, 5], [0, 6, 4, 6, 1, -2], [8, 0, 7, -1, -6, 3], [0, 0, 4, 3, -1, 5], [6, 1, 0, 0, -1, 2], [2, -3, -2, -3, 4, -2], [-8, 3, -5, -3, 3, -1], [0, 2, -5, 0, 3, -8], [-2, 0, 2, 0, -4, 4], [6, -5, -3, -2, -4, -2], [16, -2, -1, -3, 1, -2], [-6, -1, 8, 2, -5, 7], [10, -4, -2, 4, 0, 1], [7, 0, -3, -2, 4, -2], [2, -4, -2, -2, 0, 2], [-7, 2, -5, -1, 3, 1], [2, -6, -4, 3, -2, -1], [4, 3, -3, 5, -2, -1], [2, 7, 2, 1, -4, 0], [-4, 8, 7, 3, 3, 0], [0, 0, 6, -3, -3, 1], [-2, 9, 8, 3, 5, -2], [13, 2, 0, 5, 3, 0], [2, 6, -3, 4, -2, -6], [-18, 5, -3, -4, 0, 0], [10, -5, -2, 2, 1, -3], [6, 7, 0, -5, -1, -2], [13, 2, 1, -5, -1, 1], [-8, 1, -11, -7, 4, -7], [1, -1, 4, 0, 4, 0], [-4, 6, -3, -2, -2, 1], [-16, 1, 3, 2, 0, 6], [-5, -2, -7, 1, -3, -1], [2, -5, -1, -7, -7, 4], [-2, 2, 2, -2, -4, 8], [10, -5, 7, 4, 0, 7], [-4, 10, 6, 1, 2, -1], [-1, -1, -1, -2, -7, 9], [-9, -8, 6, -2, -2, 1], [4, 4, 6, 0, -3, -2], [6, -4, -1, -5, 1, 4], [8, 6, 5, 4, 3, -2], [-8, 1, 2, -3, 4, 3], [0, 12, 6, 6, -2, 0], [-2, -1, -1, 0, 6, -5], [12, 2, -3, -1, -2, -5], [20, -2, 2, 6, -2, -2], [-4, 5, 10, 8, 1, -1], [-14, 13, -4, 4, 1, -3], [-4, 2, 6, -2, 8, 0], [-14, 0, -2, -3, -2, 1], [-12, -1, -8, -9, -3, 0], [15, 5, 5, 1, -4, -4], [16, -1, 5, -6, -6, 0], [5, 4, 9, 5, -4, 3], [-8, -10, -9, -7, -3, 0], [4, 0, -2, 4, -2, 0], [-4, -6, -7, 4, 0, 1], [11, -14, 1, -4, 1, 2], [8, -7, -5, -5, 3, -7], [-4, -7, 4, -6, -1, 3], [-12, 7, -2, 5, 6, -2], [-2, 7, -1, -2, 1, 5], [18, -4, 1, -11, -5, 4], [4, 2, -4, -2, -1, 6], [8, 1, 1, 3, 6, -3], [2, -2, -13, 2, 6, -3], [-18, -4, -8, -5, 6, 2], [0, 6, 2, -6, 1, -4], [6, -3, 15, 1, -8, 9], [6, 0, 0, -2, 4, -4], [-2, 6, 6, -1, 2, 3], [12, -9, -3, 7, 7, 1], [-2, 3, 11, -4, -4, 2], [-14, 6, -6, -4, 0, -6], [-8, 4, -2, 2, 0, 4], [2, 0, 4, -4, -5, 2], [12, -18, -8, -10, -2, 0], [-7, 10, -6, -3, 3, 6], [2, -8, -13, -15, 2, -3], [-2, 0, -15, 1, 3, -5], [-8, -3, -10, -1, 0, -1], [16, 4, 19, 6, -6, 4], [2, -2, -6, -6, 0, -6], [-20, 10, -3, 2, 2, -5], [2, 8, -2, 8, 12, -2], [-4, 0, 6, -4, 2, 6], [2, 4, 9, 5, 4, 7], [-16, 6, -7, 1, 6, -8], [-4, 0, 5, -1, 1, -3], [9, 10, 19, 5, 1, -1], [-18, 4, 10, -4, -2, 8], [0, -2, 4, 0, 4, -6], [0, 9, -7, 6, 2, 0], [6, 2, 4, 12, -6, 2], [8, -2, -12, -2, 8, -14], [0, -4, -2, -2, -9, 1], [-14, -4, 3, -3, 4, -2], [-10, -4, -2, -7, 1, -5], [0, -2, 7, 0, -6, 8], [-5, 6, 1, -3, 1, -1], [-20, 4, 1, -1, -5, 4], [7, 12, 7, 5, 6, -1], [2, -7, 2, -8, -5, 5], [-2, -3, -3, -8, -6, 8], [6, -2, 11, 5, -2, -5], [-2, -2, -8, 4, 5, -2], [-30, 10, -7, -7, -3, 2], [12, 4, -2, 1, -1, 5], [6, -8, 0, -2, 0, 0], [-11, 7, 3, 2, 1, 7], [-2, -17, 3, -11, -8, -1], [20, -3, -8, -3, 9, -10], [-11, 13, 5, -3, -8, 2], [20, -8, 8, -2, -8, 2], [-12, 11, 8, 10, 5, 1], [-23, -12, -13, -3, 1, 1], [8, 10, 4, 8, -7, 3], [-18, 1, -5, 0, -12, 2], [-2, 24, 6, 10, -4, -4], [4, 6, 2, 5, 10, -1], [4, -14, -3, -5, 2, 2], [-6, 1, -2, -7, 5, -4], [-8, 2, -13, 9, 7, -5], [4, 6, -4, 4, 12, -8], [-16, -6, -8, -16, 2, -6], [-6, -13, -19, -12, 0, 5], [-8, 6, 2, -2, 13, 0], [32, 3, 12, 8, -5, 3], [-2, 2, -3, -4, -3, -2], [10, 5, 8, 5, -7, -2], [-11, -3, -5, -5, 0, 10], [17, -2, 3, 6, 1, 0], [-1, -13, -4, 4, -4, -2], [-10, 0, -4, 12, 7, 2], [5, -10, -1, -5, 3, -1], [6, -3, 2, 5, -4, -7], [-4, -1, -2, -8, -3, 10], [-8, -4, -10, -17, 2, -3], [-13, 4, -4, -16, -2, -3], [-14, 10, 4, 2, -6, -4], [24, -1, 5, 8, 1, -1], [-8, -2, -2, 0, 10, 2], [-4, 3, 0, 6, 5, -13], [13, 12, -3, 3, 3, -5], [-26, -5, -2, -2, -1, 4], [13, -10, -9, 1, 12, -5], [4, -2, -8, 10, -5, -6], [-16, 20, 3, -2, -8, 3], [-8, 6, -9, -9, 2, -3], [-14, -2, -6, -4, 12, -12], [20, -3, 15, 5, -9, 3], [10, 3, 6, -2, 7, 7], [-17, -4, -1, -7, 11, -9], [6, 4, -4, -7, -5, 3], [3, -6, 1, -6, -8, 4], [-29, -6, 1, 3, 1, 7], [6, -24, -6, -14, -2, 2], [10, 24, 14, 12, 6, -2], [7, -5, -2, 0, -4, 8], [-6, 24, 7, 5, 0, -1], [7, -8, -2, 2, -10, -7], [-16, 2, 6, -4, 0, 8], [-20, 7, -5, 3, -3, 0], [-20, 8, -8, 0, -2, -14], [-18, -5, 7, -9, -3, -4], [26, -16, -2, -4, -5, -4], [-20, 6, -1, 15, 4, -7], [-6, 6, -1, 3, 0, -5], [6, -10, -8, -6, -1, -12], [5, 25, 10, 11, 5, -3], [22, -15, -13, -13, 9, -7], [4, -2, 0, 10, 4, 0], [-4, -12, -4, -12, -3, 0], [29, 6, 6, 8, -5, 3], [-8, 7, -4, -7, 4, -4], [-10, -17, 11, -1, -11, 12], [18, 0, -2, 5, 8, -11], [18, -22, 2, 4, 6, 0], [16, -11, 5, 2, 3, 9], [22, -13, -4, 9, 10, -2], [-13, 12, -11, 11, 9, -13], [-19, 3, -6, 11, 5, -1], [-8, -13, -7, -12, -14, -4], [12, 20, 9, 5, -5, -4], [2, -22, 4, -12, -1, 5], [-36, 4, 8, 5, 1, 5], [0, 12, 6, -4, 0, 0], [18, -2, -9, -5, 5, -7], [6, -2, -3, -11, -6, -6], [-12, -8, 12, 2, -6, 0], [-16, -10, -3, -16, -4, -7], [2, 4, -9, 12, -2, -9], [-13, -10, 1, -1, -5, 5], [18, -22, -5, -9, -4, 6], [1, 14, 3, -6, -6, -4], [-4, 4, 1, -5, 10, -1], [-6, 1, -1, 13, 7, -9], [-18, 2, -2, -6, 8, -8], [0, 26, 6, 18, -1, -2], [4, 2, -4, 0, -7, 6], [-36, 24, 10, 5, -7, 5], [-22, 9, 16, 3, -3, 2], [42, 4, 1, -1, -4, 2], [26, -12, 2, -12, -16, 2], [26, -4, -5, -1, -5, -11], [12, 4, 8, 12, -9, 0], [-11, 2, 1, -11, -3, 9], [2, -25, -7, -15, 5, 3], [12, -4, 19, 8, -6, -5], [10, -6, -5, -7, 0, 11], [24, 0, 6, 10, 2, -2], [3, 0, 10, 3, -3, 12], [-2, 18, 6, 2, 4, 4], [-20, 4, 10, 6, -6, 4], [13, 6, 5, 11, -7, -5], [-16, 17, -5, -4, -6, 4], [-32, -16, -15, -6, 4, 1], [-2, 2, 10, 8, -8, 0], [18, 11, -6, -5, 6, -8], [16, -20, -7, 7, 0, -2], [-4, 12, 16, 18, 0, 8], [-22, 12, 3, 11, 7, -4], [14, -1, 2, -1, 2, 4], [26, 5, -10, -4, 5, -10], [4, -6, -5, 3, 6, -9], [-32, 15, -2, 1, 5, 8], [-12, -16, 3, -1, -10, 9], [24, 2, 2, -8, -6, -8], [0, 0, -5, 12, 0, -9], [-12, 5, 15, -3, -1, 9], [0, 4, 6, 8, -11, 16], [25, 4, -5, 5, 7, -5], [-12, 2, -6, 0, 12, -2], [-44, 8, -6, 10, 10, -6], [-30, 6, -16, 2, 4, -12], [-21, -12, -5, -9, 12, -7], [-18, -9, -14, -13, 12, 3], [13, -10, 4, -3, 1, 12], [0, -3, -2, 2, 15, -7], [-10, -2, 0, 16, 3, 4], [-38, 6, 13, 7, -3, 4], [-2, -11, -13, -3, -8, 3], [-2, -26, 1, 1, 0, 7], [20, 0, -2, -11, -6, -3], [-8, 6, -4, -2, 9, 9], [-22, -7, 14, -4, -5, 15], [-2, 6, 13, 13, 4, -5], [33, -24, 2, -8, 1, 5], [18, -8, -23, -5, 5, -12], [-12, 14, 21, 2, -8, 11], [8, -7, 1, -2, -4, 12], [44, 0, 4, 4, -4, -10], [-16, 8, -6, -1, -2, -3], [12, -23, 3, -4, -13, -1], [-24, -9, 9, 4, -6, -2], [-40, -3, -19, -12, -10, 2], [20, -13, 11, -11, -5, 21], [22, -2, -16, -4, 6, -14], [8, -3, -18, -9, 11, -10], [-2, 0, -8, 11, -6, 2], [16, 15, -8, 5, -3, -2], [-4, -1, -5, 6, -6, 0], [14, 0, -10, 9, -8, -7], [0, 2, -2, -8, -8, 0], [8, 24, 11, 5, 2, -15], [24, 8, 31, 11, -3, 10], [-24, 9, 6, 21, 8, -7], [56, -8, 7, 3, 5, 2], [3, -8, 11, -3, 7, 9], [13, 7, 7, 7, 2, 4], [-18, 7, -1, -14, -9, -1], [-16, 0, -3, 5, -4, 6], [-13, 16, 9, 0, -5, -4], [-22, 16, 0, -2, 2, -2], [-42, -3, -4, 6, 13, -6], [4, -12, -20, -11, -8, 3], [-6, -24, -4, -10, -2, 9], [11, 16, 3, 0, 11, -6], [38, -21, -16, -21, 2, -2], [-22, -4, -10, -6, 0, -6], [-16, 6, -19, -9, 6, 1], [5, -26, -5, -18, -4, 4], [-14, -8, 19, -4, -10, 5], [50, -14, 1, -19, -7, 11], [6, 2, 2, -9, -19, 15], [1, -7, -20, -21, 1, -9], [-15, -8, 8, -6, 3, -7], [-8, 20, -1, 2, -16, 9], [-1, 0, 8, -8, -18, 1], [-23, -1, 10, 4, 3, 0], [9, 7, 15, 6, 0, 5], [-8, 17, 10, -2, -5, 7], [22, 23, 15, -2, -8, -4], [34, -13, 12, 5, 8, 13], [-18, -11, -15, 7, 8, -3], [7, 12, -21, -5, 11, -13], [-36, 4, -12, -10, 4, 4], [-10, -23, 10, -6, -11, 7], [36, -12, -10, 0, 1, -13], [20, -7, 31, -3, -15, 12], [-14, 9, 19, -4, -16, 12], [-20, 13, 2, -3, -16, 0], [18, 12, 7, 3, 4, 6], [-12, -15, -16, -10, 4, 10], [15, 20, 26, 12, 9, -1], [51, -6, 25, 13, -3, 9], [4, -2, -13, 12, 2, -13], [-17, -2, 1, 11, -3, 11], [33, 3, -4, -16, 3, -6], [18, -20, 4, -18, -5, 4], [-18, -20, 9, -11, -13, 6], [2, 13, 5, 12, 2, 2], [-8, 6, 8, 12, 2, -6], [0, 18, 9, -8, -4, -3], [-12, 2, -20, -14, 0, 0], [-16, 22, 2, 13, 0, 2], [-10, 21, -2, 6, 16, 0], [-2, -4, -26, 10, 4, -10], [-4, 3, -17, 10, 13, -1], [-42, 12, -17, -1, 10, -1], [12, 0, 14, 14, 4, 12], [-38, -10, 0, 4, -14, 10], [38, -30, -1, -17, -7, 14], [-6, -11, -14, -9, 2, 2], [-33, 2, -9, -13, -11, -13], [-8, 22, 6, 24, 11, -4], [-4, 16, 0, 4, 2, -6], [-16, -2, 11, 0, -12, 23], [42, 17, 19, 5, 3, -6], [-4, 11, -12, -4, 13, 3], [-19, -2, 9, 5, 11, -13], [22, 14, 14, 18, -19, 6], [-8, -20, -15, -7, -7, 17], [13, 0, 1, 7, 1, -3], [-2, -4, 4, 14, -2, -14], [-18, 7, 24, 5, -7, 2], [12, 0, -20, -10, 7, -1], [-26, 10, -8, 24, 23, -10], [-6, 11, 13, -3, 3, -1], [-4, 21, -1, 15, -8, 3], [24, -23, -3, -1, 13, -11], [18, -20, -1, -11, 1, -5], [20, -6, -14, 1, 13, -13], [8, 15, 22, 25, -18, 13], [-4, 1, 15, 2, -20, 18], [34, -5, 10, -6, -21, 15], [3, -14, -15, 5, -5, -5], [16, 0, 0, 18, 4, 12], [-24, 11, 4, -3, -6, 10], [22, 31, 22, 12, -3, -13], [-41, -8, -1, 15, 2, 1], [29, -1, 22, -1, -19, 15], [-21, 9, -23, 9, 18, -12], [40, -6, 13, 3, -6, 0], [2, -4, -10, -8, 1, 1], [60, 3, 2, 7, -5, -2], [40, 10, 8, 14, 6, 0], [4, -2, -18, 2, 8, -6], [8, 26, -8, 7, 23, -9], [25, -9, -3, 7, -8, -10], [-4, 4, -6, -4, -11, -16], [-6, 3, -3, 9, 4, 17], [27, 9, 21, 6, -7, 5], [6, -16, 7, 21, 2, 3], [5, -6, 19, 4, 0, 10], [-6, -23, -9, 4, 6, 0], [-20, -5, 1, 9, 7, 1], [0, 2, 13, 12, 8, 3], [8, 9, -21, -3, 3, 5], [1, 13, 8, -3, 1, -27], [22, 11, 0, -2, -5, 3], [-34, -4, 8, 12, -3, 14], [-22, 11, -1, 17, 3, -7], [21, -2, -20, -8, 11, -11], [16, 27, -12, 4, 15, -17], [-32, 1, -27, 3, 4, -15], [-16, -8, -13, -17, -11, -5], [30, 10, 19, 18, -6, 9], [-4, 26, 12, 28, 2, 6]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_7448_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_7448_2_a_bn();". // 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_7448_2_a_bn(:prec:=6) chi := MakeCharacter_7448_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_7448_2_a_bn();". // 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_7448_2_a_bn( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7448_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![7, -19, 2, 19, -6, -3, 1]>,<5,R![53, -123, 59, 24, -17, -1, 1]>,<11,R![575, -539, 4, 121, -28, -3, 1]>,<13,R![-3920, -1544, 639, 194, -40, -6, 1]>,<17,R![148, 322, 91, -138, -55, 2, 1]>],Snew); return Vf; end function;