
// Make newform 2548.2.j.q in Magma, downloaded from the LMFDB on 05 January 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2548_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2548_j_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_2548_2_j_q();".
// 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_2548_2_j_q();".
// 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 := [4, 0, 32, 0, 36, 0, 11, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [2, 10, 0, 7, 0, 1, 0, 0], [3, 0, 1, 0, 0, 0, 0, 0], [-4, 0, -6, 0, -1, 0, 0, 0], [6, 30, 2, 31, 0, 10, 0, 1], [-10, 0, -22, 0, -9, 0, -1, 0], [-4, 18, -6, 28, -1, 10, 0, 1], [-10, 40, -22, 63, -9, 21, -1, 2]];
    Rf_basisdens := [1, 4, 1, 1, 4, 2, 2, 4];
    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_2548_j();"
function MakeCharacter_2548_j()
    N := 2548;
    order := 3;
    char_gens := [1275, 885, 197];
    v := [3, 2, 3];
    // 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_2548_j_Hecke();"
function MakeCharacter_2548_j_Hecke(Kf)
    N := 2548;
    order := 3;
    char_gens := [1275, 885, 197];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 0, 0, 0, 0, 0, 0], [1, 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<X, Y> := 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<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..prec]];
    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, 1, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 1, -1, -1, -1, 1], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, -1, -2, -1, 2], [0, 0, 0, 0, 0, 0, 0, -1], [-3, 3, 1, 0, -1, 0, 0, 0], [2, 0, 1, 2, 0, -2, 0, 0], [0, -1, 0, 0, -1, 2, 0, -2], [-7, 7, 1, 0, -1, 0, 0, 0], [-1, 0, -1, -1, 0, 0, 0, 0], [1, 0, 1, -1, 0, 0, 0, 0], [1, -1, -1, 0, 1, 0, 1, 1], [0, -5, 0, 0, 2, 2, 0, -2], [0, -3, 0, -1, -3, 1, 1, -1], [3, -3, -2, 0, 2, 0, 0, -4], [0, 2, 0, -2, -4, -1, 2, 1], [-2, 0, 0, 3, 0, -5, 0, 0], [0, -2, 0, -1, 2, -2, 1, 2], [-4, 4, -2, 0, 2, 0, 1, -4], [3, 0, -3, -3, 0, 0, 0, 0], [-7, 7, 5, 0, -5, 0, 2, 4], [-1, 0, -3, -1, 0, 0, 0, 0], [0, -2, 0, -3, 0, 0, 3, 0], [-7, 7, -1, 0, 1, 0, 0, 2], [2, -2, -4, 0, 4, 0, -3, -2], [0, -3, 0, -4, -3, 0, 4, 0], [10, 0, -1, 2, 0, 2, 0, 0], [8, 0, -4, -6, 0, -2, 0, 0], [8, -8, -2, 0, 2, 0, -1, 0], [0, -10, 0, 1, 0, 2, -1, -2], [0, 0, -4, 0, 0, -4, 0, 0], [-8, 8, -4, 0, 4, 0, 3, -2], [0, -8, 0, -2, 4, 1, 2, -1], [0, 4, 0, -1, 5, 4, 1, -4], [0, 0, -2, 0, 2, 0, 0, 5], [0, 0, -6, -3, 0, 3, 0, 0], [1, -1, 2, 0, -2, 0, 2, -4], [0, 1, 0, 2, -3, 0, -2, 0], [-4, 0, 1, 6, 0, 0, 0, 0], [0, 0, 4, 0, -4, 0, 1, 4], [0, 13, 0, 0, -3, 0, 0, 0], [5, 0, 5, -1, 0, 2, 0, 0], [0, 1, 0, 2, -1, -2, -2, 2], [-5, 0, -5, -3, 0, 0, 0, 0], [7, 0, -7, 0, 0, -7, 0, 0], [0, -9, 0, 0, 3, -2, 0, 2], [-11, 11, -7, 0, 7, 0, 0, -2], [-11, 11, 0, 0, 0, 0, -2, -6], [3, 0, -1, 4, 0, 5, 0, 0], [0, 3, 0, 2, 1, -6, -2, 6], [3, 0, 1, -1, 0, 2, 0, 0], [0, 0, 6, 0, -6, 0, 3, 4], [0, -15, 0, 2, 7, 8, -2, -8], [0, 10, 0, 5, 3, 0, -5, 0], [16, -16, 6, 0, -6, 0, 6, -3], [0, 7, 0, 2, -6, -4, -2, 4], [-1, 0, -7, -3, 0, -2, 0, 0], [0, -2, 0, -7, -2, 2, 7, -2], [1, 0, 5, -1, 0, -2, 0, 0], [1, 0, -3, 0, 0, 3, 0, 0], [0, -15, 0, 0, 3, 4, 0, -4], [14, -14, 2, 0, -2, 0, -5, 2], [-4, 4, 0, 0, 0, 0, -5, -2], [1, -1, 5, 0, -5, 0, -2, -2], [2, 0, -1, 4, 0, -8, 0, 0], [0, 1, 0, -2, 7, 0, 2, 0], [7, 0, 5, -1, 0, 4, 0, 0], [0, 11, 0, -2, -5, 2, 2, -2], [-7, 7, 5, 0, -5, 0, 2, 6], [0, -8, 0, 3, 4, -2, -3, 2], [4, -4, 3, 0, -3, 0, -3, 4], [-13, 0, -1, 3, 0, -10, 0, 0], [-1, 1, 5, 0, -5, 0, -4, 12], [0, 0, 0, -1, -5, -10, 1, 10], [-10, 10, 6, 0, -6, 0, 3, -2], [1, -1, 5, 0, -5, 0, -6, 2], [0, 6, 0, 3, -8, -2, -3, 2], [6, 0, -8, 2, 0, -4, 0, 0], [-6, 0, -4, -4, 0, 4, 0, 0], [0, 1, 0, -2, -5, 0, 2, 0], [-1, 0, 4, -1, 0, 0, 0, 0], [6, -6, 0, 0, 0, 0, 5, 0], [10, -10, 10, 0, -10, 0, 7, -4], [14, 0, 2, 0, 0, 0, 0, 0], [-5, 5, 15, 0, -15, 0, 4, 2], [21, 0, 1, -7, 0, 4, 0, 0], [-17, 0, -3, -3, 0, 0, 0, 0], [2, -2, 0, 0, 0, 0, -3, -8], [0, 19, 0, 3, -1, -5, -3, 5], [0, -1, 0, -9, -7, 1, 9, -1], [0, 0, -6, -8, 0, 0, 0, 0], [1, -1, 5, 0, -5, 0, 2, 2], [3, 0, -3, 5, 0, -12, 0, 0], [15, -15, -1, 0, 1, 0, -4, 0], [0, -6, 0, 15, 6, 2, -15, -2], [-7, 7, -7, 0, 7, 0, 2, -4], [14, -14, 2, 0, -2, 0, 3, -8], [3, 0, 9, 7, 0, 2, 0, 0], [0, -6, 0, 7, 8, -4, -7, 4], [0, -8, 0, 3, 10, 2, -3, -2], [3, -3, -2, 0, 2, 0, 4, 2], [0, 3, 0, 2, 7, 6, -2, -6], [0, -1, 0, -4, 5, 12, 4, -12], [16, 0, 2, 1, 0, 7, 0, 0], [6, -6, -14, 0, 14, 0, 1, -8], [0, 2, 0, -11, -10, 0, 11, 0], [18, 0, -7, -6, 0, -2, 0, 0], [14, -14, 10, 0, -10, 0, 3, 2], [0, -6, 0, 11, 2, 0, -11, 0], [-3, 0, 5, -1, 0, -2, 0, 0], [0, 7, 0, -6, -3, 0, 6, 0], [-9, 0, -3, -11, 0, 0, 0, 0], [0, -8, 0, -5, -12, 2, 5, -2], [-4, 0, 6, -7, 0, 11, 0, 0], [0, -8, 0, 5, 10, 4, -5, -4], [-12, 12, 8, 0, -8, 0, 7, 2], [16, 0, -4, 0, 0, -4, 0, 0], [0, 5, 0, 10, 5, -4, -10, 4], [2, 0, 6, 14, 0, 6, 0, 0], [11, -11, 10, 0, -10, 0, 0, 0], [0, 13, 0, 2, -13, -4, -2, 4], [-13, 13, 7, 0, -7, 0, 1, 11], [-8, 0, -6, -10, 0, 8, 0, 0], [-4, 4, -8, 0, 8, 0, 7, -6], [15, -15, 3, 0, -3, 0, -4, 0], [19, 0, -7, -7, 0, 0, 0, 0], [-5, 5, -3, 0, 3, 0, 4, 0], [0, 9, 0, 0, 9, 2, 0, -2], [25, 0, -5, -4, 0, 3, 0, 0], [9, -9, 1, 0, -1, 0, 2, -10], [-5, 0, 4, 1, 0, 2, 0, 0], [-2, 2, 10, 0, -10, 0, 1, 8], [-16, 0, 10, 6, 0, 4, 0, 0], [0, -13, 0, -2, -13, -2, 2, 2], [0, 8, 0, 0, 12, 9, 0, -9], [-10, 0, 6, 10, 0, -2, 0, 0], [0, -14, 0, 5, -3, -10, -5, 10], [-5, 0, 1, 5, 0, 0, 0, 0], [-4, 4, 6, 0, -6, 0, -1, -10], [0, 17, 0, 2, -1, 0, -2, 0], [-6, 0, -10, -11, 0, 1, 0, 0], [0, 15, 0, -2, -6, 10, 2, -10], [29, 0, -3, 2, 0, -7, 0, 0], [6, 0, -2, -2, 0, -6, 0, 0], [0, -15, 0, -6, 4, 12, 6, -12], [8, -8, 2, 0, -2, 0, 13, -2], [-1, 1, -9, 0, 9, 0, 0, -10], [3, -3, -13, 0, 13, 0, -2, -4], [-16, 0, -12, 4, 0, -6, 0, 0], [-13, 0, -13, -9, 0, 6, 0, 0], [11, -11, 3, 0, -3, 0, -4, 0], [0, 6, 0, -5, 14, 10, 5, -10], [7, 0, 7, 1, 0, 4, 0, 0], [9, -9, 1, 0, -1, 0, -16, 8], [-5, 5, -11, 0, 11, 0, -8, -4], [15, 0, -6, -9, 0, 4, 0, 0], [0, -7, 0, -14, -11, 0, 14, 0], [13, -13, 3, 0, -3, 0, -1, 5], [29, 0, -2, -13, 0, 2, 0, 0], [-8, 0, 8, -1, 0, 5, 0, 0], [20, -20, 0, 0, 0, 0, 7, -4], [0, -24, 0, 1, 2, 0, -1, 0], [0, -12, 0, 12, 2, 1, -12, -1], [0, 7, 0, -12, -7, -8, 12, 8], [0, 11, 0, -10, 4, 2, 10, -2], [-12, 0, 0, 2, 0, 8, 0, 0], [-21, 21, -14, 0, 14, 0, -2, -2], [0, -22, 0, 8, 14, 5, -8, -5], [-4, 0, 2, 8, 0, -6, 0, 0], [11, -11, -9, 0, 9, 0, 6, 4], [0, 17, 0, -6, -9, -2, 6, 2], [0, -45, 0, 2, 1, -4, -2, 4], [7, 0, 6, 7, 0, 0, 0, 0], [-1, 0, 11, -8, 0, 17, 0, 0], [0, -35, 0, -6, -5, 6, 6, -6], [-10, 0, 12, 0, 0, -6, 0, 0], [-18, 18, -7, 0, 7, 0, 3, -10], [-33, 33, -1, 0, 1, 0, 1, -7], [-25, 0, 1, -1, 0, -14, 0, 0], [-13, 0, 16, 1, 0, -2, 0, 0], [-24, 24, 12, 0, -12, 0, 11, 8], [0, -10, 0, -2, -4, -1, 2, 1], [0, -4, 0, 15, 2, -12, -15, 12], [0, -10, 0, 5, 1, 2, -5, -2], [0, -15, 0, 8, -5, -12, -8, 12], [3, -3, 7, 0, -7, 0, 6, 12], [0, -32, 0, 4, -2, -3, -4, 3], [-2, 2, 11, 0, -11, 0, 7, 14], [1, 0, -1, 6, 0, -3, 0, 0], [15, -15, -5, 0, 5, 0, 10, 0], [40, -40, 6, 0, -6, 0, -5, 4], [0, 0, 0, 11, 6, -12, -11, 12], [0, 21, 0, 0, -18, -2, 0, 2], [0, -4, 0, 9, -16, -12, -9, 12], [-16, 16, 12, 0, -12, 0, -9, 2], [-34, 0, 18, 2, 0, 10, 0, 0], [42, -42, -4, 0, 4, 0, -7, 4], [0, -15, 0, -6, 3, -2, 6, 2], [-19, 0, 3, -13, 0, -4, 0, 0], [9, -9, 7, 0, -7, 0, -8, 14], [0, 28, 0, -5, 2, 16, 5, -16], [19, 0, -13, -13, 0, -6, 0, 0], [0, 24, 0, -5, -10, -12, 5, 12], [-18, 18, -2, 0, 2, 0, 12, -7], [-2, 2, 4, 0, -4, 0, 7, -8], [2, 0, -8, 0, 0, -8, 0, 0], [0, -6, 0, -3, -6, -2, 3, 2], [-11, 11, 0, 0, 0, 0, -6, -2], [-35, 0, -6, -7, 0, 4, 0, 0], [16, 0, 18, -2, 0, 2, 0, 0], [-18, 18, -4, 0, 4, 0, -10, 15], [0, 15, 0, -16, -17, -6, 16, 6], [-21, 21, -1, 0, 1, 0, -10, 2], [0, -18, 0, -5, 16, 10, 5, -10], [0, -17, 0, -6, -14, 2, 6, -2], [-12, 12, -6, 0, 6, 0, -4, -7], [-21, 0, -7, 7, 0, 0, 0, 0], [9, -9, 4, 0, -4, 0, -2, -2], [7, 0, -11, -7, 0, 8, 0, 0], [-19, 19, 9, 0, -9, 0, -2, -2], [-29, 29, 3, 0, -3, 0, -8, 6], [37, 0, 1, 7, 0, -4, 0, 0], [-19, 0, -2, 5, 0, -12, 0, 0], [20, -20, 4, 0, -4, 0, -7, 18], [0, -21, 0, 16, 5, 0, -16, 0], [-14, 14, -16, 0, 16, 0, -11, -4], [-14, 14, -14, 0, 14, 0, -8, 3], [0, -2, 0, -1, -13, -12, 1, 12], [0, -3, 0, 10, 3, -14, -10, 14], [-6, 0, -20, -1, 0, 3, 0, 0], [0, -18, 0, 3, -7, -8, -3, 8], [-40, 0, 4, -2, 0, 6, 0, 0], [0, -7, 0, -18, -3, -10, 18, 10], [3, -3, -19, 0, 19, 0, -4, 4], [-16, 16, -4, 0, 4, 0, -5, -10], [-5, 0, 7, 13, 0, 0, 0, 0], [-3, 0, 5, -1, 0, -4, 0, 0], [0, -13, 0, -7, 5, 7, 7, -7], [-29, 29, 9, 0, -9, 0, 0, 12], [0, 3, 0, 4, -1, -10, -4, 10], [6, -6, 2, 0, -2, 0, -7, 6], [4, 0, -10, 16, 0, -6, 0, 0], [8, -8, -8, 0, 8, 0, -15, 6], [-24, 0, 10, 21, 0, -1, 0, 0], [0, 7, 0, -9, 9, 3, 9, -3], [0, 0, 0, -16, 2, -5, 16, 5], [31, 0, 3, 11, 0, 6, 0, 0], [16, 0, 10, -12, 0, 14, 0, 0], [-16, 16, -2, 0, 2, 0, -7, -6], [0, 1, 0, -11, -15, -3, 11, 3], [2, 0, -5, -14, 0, 22, 0, 0], [0, -6, 0, 1, 15, -4, -1, 4], [21, -21, -15, 0, 15, 0, 2, -14], [0, 18, 0, 1, -13, -18, -1, 18], [0, 41, 0, -3, -3, 3, 3, -3], [0, 0, -7, -14, 0, 12, 0, 0], [-10, 10, -14, 0, 14, 0, -17, 10], [0, 5, 0, 12, -9, -6, -12, 6], [2, 0, 10, 12, 0, -10, 0, 0], [0, 9, 0, 0, 5, 6, 0, -6], [-18, 0, 32, 0, 0, 8, 0, 0], [0, -9, 0, -6, 3, -14, 6, 14], [20, -20, 0, 0, 0, 0, -19, 2], [-8, 0, 0, 6, 0, 12, 0, 0], [18, 0, -16, -4, 0, -20, 0, 0], [-39, 0, 7, 11, 0, -4, 0, 0], [0, -17, 0, -18, 2, -4, 18, 4], [18, -18, 16, 0, -16, 0, 9, -12], [0, 45, 0, -7, -3, -7, 7, 7], [0, 20, 0, 17, 2, -6, -17, 6], [1, -1, -15, 0, 15, 0, 0, 0], [18, 0, -16, -4, 0, -20, 0, 0], [20, -20, -12, 0, 12, 0, -2, -15], [44, -44, -22, 0, 22, 0, -6, -17], [0, 6, 0, 7, 2, 12, -7, -12], [19, -19, 5, 0, -5, 0, 2, 4], [-19, 19, -11, 0, 11, 0, 12, -10], [0, 26, 0, -23, 2, 10, 23, -10], [0, -5, 0, 6, 11, -2, -6, 2], [10, 0, 16, 14, 0, -8, 0, 0], [13, 0, -25, -11, 0, -8, 0, 0], [-4, 4, -14, 0, 14, 0, -12, -1], [14, -14, 4, 0, -4, 0, -7, 6], [0, -27, 0, -4, 8, 16, 4, -16], [-1, 0, 5, 19, 0, -12, 0, 0], [0, -4, 0, 0, 6, 1, 0, -1], [-23, 0, -4, -5, 0, -6, 0, 0], [0, -17, 0, 16, 22, 10, -16, -10], [0, 40, 0, 11, 2, 6, -11, -6], [-12, 12, 6, 0, -6, 0, 9, 8], [-1, 0, 9, -4, 0, 5, 0, 0], [8, 0, 26, -6, 0, 8, 0, 0], [0, 12, 0, 5, -11, -22, -5, 22], [4, -4, 0, 0, 0, 0, -1, 8], [-32, 0, -13, 4, 0, -14, 0, 0], [33, -33, -5, 0, 5, 0, 12, 6], [-1, 0, -11, 1, 0, 4, 0, 0], [21, -21, 6, 0, -6, 0, -4, 22], [-32, 32, 24, 0, -24, 0, -3, 4], [0, 3, 0, -14, 11, 2, 14, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2548_j_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2548_2_j_q();".
// 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_2548_2_j_q(:prec:=8)
    chi := MakeCharacter_2548_j();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_2548_2_j_q();".
// 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_2548_2_j_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2548_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![144, 144, 216, 0, 60, -6, 15, -3, 1]>,<5,R![256, -128, 256, 64, 136, 4, 13, 1, 1]>],Snew);
    return Vf;
end function;