
// Make newform 6004.2.a.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6004_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_6004_2_a_d();".
// 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_6004_2_a_d();".
// 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 := [248, 340, -241, -248, 87, 56, -15, -4, 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], [-56, -68, 37, 36, -10, -4, 1, 0], [72, 140, 31, -73, -26, 14, 3, -1], [64, 4, -104, 25, 40, -10, -4, 1], [112, 32, -171, 7, 56, -8, -5, 1], [128, 92, -200, -29, 66, -4, -6, 1], [160, 188, -229, -69, 76, 0, -7, 1]];
    Rf_basisdens := [1, 1, 8, 16, 8, 8, 8, 8];
    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_6004_a();"
function MakeCharacter_6004_a()
    N := 6004;
    order := 1;
    char_gens := [3003, 2529, 3953];
    v := [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_6004_a_Hecke(Kf)
    return MakeCharacter_6004_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<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+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, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [-1, 1, -2, -1, -1, 1, 0, -1], [-2, 1, -1, 0, -1, 0, -1, 0], [-1, 0, 0, 0, 0, -1, 1, 0], [1, 0, 0, 0, 0, 0, 0, 0], [2, 0, 1, -1, 0, 0, 0, 1], [0, 0, 2, 0, 1, -1, 1, -1], [-1, 2, -3, 0, -1, 1, 0, -2], [-1, 1, -1, -1, 0, 0, -1, 2], [-1, 1, 1, -1, 0, -2, -1, 0], [1, 0, -1, 1, 1, 0, 0, 0], [2, -2, -1, 2, -1, 1, -1, -1], [0, -1, 2, 3, 0, 1, 0, 0], [0, 2, 0, -2, -2, 0, 0, -2], [1, 2, -1, -1, 0, 1, 1, -1], [3, -1, 0, 0, 0, -1, 1, 0], [6, -1, 0, 1, -1, 2, -1, 1], [-3, -1, 2, 1, 2, -1, 1, -1], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, -1, 0, -2, 0, 1], [-2, -1, 3, 2, 3, -2, 1, 0], [-4, 0, -2, 2, 0, 2, 0, -2], [2, 4, -3, -1, -1, -2, -1, 1], [2, -3, 4, -1, 1, -4, 3, 1], [6, -1, -4, 3, -1, 4, -1, 1], [1, 4, 0, -2, 1, -1, 1, 0], [3, 1, -2, 1, 0, 1, 0, 3], [8, 0, 0, -2, 0, 2, 2, 0], [1, -3, 4, -1, 3, -3, 0, 1], [3, 0, 1, 1, 2, 1, 3, -1], [2, -2, 2, 1, 1, 1, -1, -2], [7, 2, -1, -3, 0, -1, -3, 1], [0, -2, 0, 0, -1, 0, 3, 0], [-2, -2, 1, 5, 2, 0, 0, 1], [6, -6, 0, 2, 0, 4, 0, 0], [6, -2, 0, -2, 1, -4, -3, 4], [1, -1, 1, 3, 0, 2, -3, 2], [4, -2, -2, 2, -1, 4, -1, -2], [0, 2, -4, -2, 0, 2, 2, -2], [1, 0, 1, -1, -1, -6, 2, 0], [10, 0, 4, 0, 1, -5, -1, 5], [0, 4, -4, -4, -2, 2, -2, -4], [-11, 4, -2, -2, -1, -2, 0, -1], [2, 1, -2, -3, 1, -2, 3, -3], [-4, 2, -4, -6, -1, -2, 1, 0], [4, -1, -2, 1, -3, 2, -3, -1], [7, -6, -3, 5, 2, 1, 1, -1], [-3, -2, 4, 0, 0, -3, 1, 0], [7, 3, -5, -2, -3, 5, 0, -2], [6, -5, 3, 2, 1, -2, 3, 0], [0, 2, -6, -2, 1, 0, -2, -2], [2, 4, -4, 0, -4, 0, -2, 0], [-5, -3, 1, 2, -1, -1, 4, -4], [-2, -7, 3, 6, 3, 0, 5, -2], [-5, -2, 1, 0, 1, 4, 0, -3], [11, -5, -2, 1, 5, -1, 2, -1], [-4, 3, -3, -2, -5, 6, -1, -2], [-1, 1, 2, -1, -3, 1, -2, -1], [1, -4, -4, 2, -1, 3, 1, 0], [2, 4, -4, 0, -6, 4, 2, -6], [-8, 4, -3, -1, -1, -4, 4, -5], [5, 2, 4, -1, 1, -7, -2, 1], [-4, -1, 7, 0, 1, 0, -3, 6], [-6, 7, 4, -1, 3, -4, 1, 1], [2, 3, 1, -2, -5, 6, -1, -4], [5, 1, 5, 0, 1, -9, 0, 0], [1, -2, 2, -2, 0, 3, -1, -4], [16, 2, 3, -7, -2, -4, -2, 3], [-2, -10, 4, 3, 3, 1, 1, 4], [6, -6, -1, 3, 6, 2, -2, 3], [-7, 1, 6, -1, 0, 3, 2, -3], [6, -2, 0, 2, 2, -4, 2, -4], [0, 4, -2, -2, 0, 0, -4, -2], [7, 5, -6, -3, 2, -1, -3, 1], [6, 1, 1, 3, -4, 0, -2, 1], [0, -1, -2, -3, 2, -3, 2, 2], [-5, 3, -8, 1, 0, -3, -2, -1], [3, -2, -2, -1, -5, 6, 0, -2], [8, 1, 9, -2, -1, -4, -5, 4], [0, -9, -1, 4, 2, 2, 2, -2], [-4, -3, 5, 0, 7, -4, 3, 6], [-1, -2, -7, -4, -5, 3, -2, 0], [4, 0, 3, 1, 3, 0, -4, 9], [1, 6, -5, 0, -1, 0, -4, 5], [6, -1, -2, 0, -2, 4, -2, -2], [7, 2, -3, 1, 0, 5, 3, -1], [-11, 2, -1, 1, -3, 4, 0, 0], [-1, -7, 3, 2, 1, 3, 0, 4], [4, 0, -1, 3, 0, 4, -2, -1], [10, -5, 3, 0, 5, -6, 5, 0], [-7, 0, -8, 2, 3, 3, -1, -2], [-7, 3, 0, -1, -5, 3, -6, 3], [-2, -4, 12, -2, 3, -5, -1, 5], [3, 5, -2, 1, 2, 5, -4, 3], [8, -2, -4, 2, -3, -1, 5, -3], [7, 1, 4, 2, 2, -1, -1, 6], [-13, 5, 1, 2, -2, 1, 1, 0], [-6, 7, -3, 0, -2, 2, -2, -4], [5, -1, -1, 4, -2, 9, -3, 4], [7, -1, 4, 2, -2, -3, 1, -4], [16, -1, 9, -6, 3, -12, 3, 4], [4, 8, -5, -5, -6, 8, 0, -5], [19, -8, 5, 5, 2, 1, 1, 3], [3, -6, 3, 3, -1, -2, -4, 10], [4, 4, -5, 1, 0, -2, -2, -5], [-2, -10, 2, 4, 1, 4, -3, -6], [-7, -6, 5, 6, 1, 0, 2, 3], [2, -1, 2, -1, 3, -6, -3, -1], [13, -2, 4, 2, 0, 1, 1, 2], [1, 3, 8, -1, -6, -1, -3, 1], [4, -2, -7, 5, -1, 1, -1, -8], [6, -10, 2, 6, 1, 6, 0, 6], [4, -4, -2, 8, -2, 4, -2, 2], [-5, -3, 15, 6, 9, -11, 2, 2], [4, -10, -2, 11, -3, 9, 1, -2], [12, 3, 0, -6, -2, -6, 4, 0], [-6, 6, 4, -2, 4, -10, 0, 2], [-1, 4, 3, -6, 1, -2, 4, -9], [-11, 6, -12, 0, -9, 15, -1, -6], [14, -3, -1, -4, -1, 0, -7, 4], [4, -3, 1, 4, 2, 2, -2, -2], [2, -2, 7, 2, 1, -3, -3, 11], [2, -6, 4, 2, 2, -4, -2, -2], [-6, 6, 4, -8, 4, -10, -2, -2], [7, -7, -4, 1, 1, 7, -2, -1], [11, 3, 2, -3, 1, -9, 0, 3], [14, 8, 1, -11, 1, -8, -1, 5], [-27, -4, -4, 4, 1, 4, 6, -5], [0, -1, -1, 2, 8, 0, 2, 6], [0, 5, -7, -6, -1, 4, -7, 4], [-13, 7, -4, -5, -2, 1, 3, -11], [2, 5, -7, -1, 0, 8, -10, 5], [-3, -6, 8, 4, 8, -11, 5, 4], [-8, 4, 8, -6, 4, -10, 4, -2], [-2, 4, 8, 0, 0, 2, -2, 4], [0, 5, -6, -3, -2, 1, -2, -6], [9, -5, 1, 0, -5, 9, -4, 2], [8, -14, 4, 8, 10, 0, 6, 2], [19, -11, 6, 7, 7, -1, 4, 5], [0, -4, 6, 5, -1, -3, -1, 0], [-12, 3, 6, -5, 3, -2, 3, 5], [-8, -6, 2, 2, 7, -1, -1, 9], [-6, 1, -1, 8, 0, 2, 4, 0], [-8, -10, 8, 8, 2, -4, -4, 6], [-12, 4, -10, -6, -2, 2, -6, -2], [-12, 0, -10, 10, 3, 12, -4, -2], [4, -4, 0, 2, -4, 0, 8, -8], [14, -5, -1, 2, 3, 4, 7, -2], [-5, 10, -5, -7, -6, 9, -1, -9], [-28, 6, -2, -1, -3, 9, -3, -2], [21, -1, 2, -2, 8, -5, 1, 4], [-12, -3, -7, 0, -2, 2, 2, -6], [7, -2, 5, 0, 7, 1, 8, -2], [6, 0, -5, 3, -8, 14, -10, -1], [12, -4, 2, 0, 10, -2, 8, -4], [1, 8, 1, -5, 2, -9, 1, 5], [14, -1, -5, -10, 1, 4, 1, -6], [-11, -4, -1, -1, 5, -1, 7, -5], [0, -1, -15, -6, 1, 4, 5, -8], [-6, 4, -7, -11, 2, 4, 0, -5], [2, 1, 1, 2, 7, -2, -1, 4], [3, -10, 8, 0, 3, -11, 1, 6], [-6, 0, 0, 2, 2, 6, 6, -10], [12, 1, -4, -1, -3, 4, -9, 7], [-5, 7, -5, -2, -1, -1, -6, 8], [15, -7, 9, -5, 2, -14, 1, 8], [-27, 5, 2, 5, 2, -7, 3, 1], [0, 7, 4, -15, 7, -16, 7, 1], [-4, 16, 0, -16, -4, -8, 0, -2], [12, 12, -13, -1, -8, 6, -6, -7], [8, -4, 14, -2, 4, -8, 0, 8], [-20, 10, -10, -6, -6, -2, -2, -4], [-9, -13, 4, 9, 2, 11, 1, -3], [14, -2, 1, -1, 7, -11, 9, 0], [13, -8, 10, -2, 4, -11, -3, 2], [-10, 8, -6, -12, 0, -2, -6, -6], [-6, 1, 0, 0, 8, 0, 0, 2], [-1, 2, 6, 4, 9, -8, 4, 1], [-8, -2, -4, 4, -1, 8, -5, 8], [-11, 2, -1, -4, -3, 4, -2, 5], [-13, -3, 6, 7, 6, -15, 8, 1], [-1, 3, -1, 0, 3, -1, 8, -2], [9, -6, -4, 4, 4, -5, 3, -2], [-8, 5, 11, 2, 1, 2, -1, 2], [18, -9, 10, -3, 5, -6, -5, 5], [4, -11, 7, 10, 5, 2, 3, -2], [-8, 14, -5, -9, -4, -10, 4, -3], [1, -6, 10, 0, 1, -13, 7, 2], [-26, 2, 4, 3, -3, -3, -1, 0], [-4, -3, -3, 2, 3, 4, 1, 8], [15, -6, 0, 2, -1, 9, 1, 4], [0, -4, 1, 5, 0, -4, -2, 9], [-6, -4, -10, 6, -2, 12, 2, -6], [-24, 19, 0, -8, -8, 0, 0, 0], [16, -6, 0, 4, -4, 8, -10, 6], [17, 0, 12, -1, 5, -9, 4, 7], [-5, -3, -2, -1, -1, 3, 2, 1], [16, -10, -2, 14, -2, 14, -6, 6], [4, 0, -2, -4, 4, -4, 0, 2], [0, 4, -4, 2, -6, 10, -2, -6], [-8, -2, 8, 8, 8, -8, 4, 6], [-13, 8, -4, -12, -1, -3, 1, 0], [-7, 7, -13, -8, -2, -3, 1, 0], [-3, 6, -6, -6, -1, -2, 4, -1], [-12, 3, -3, 2, 1, 0, -9, -2], [14, 9, -18, -10, -4, 10, -10, 2], [6, -6, -12, 8, 2, 8, 2, -2], [-8, -8, 2, 16, 6, -2, 8, -4], [14, -18, 10, 8, 6, 2, 10, 2], [4, -8, 8, -5, -1, -7, -1, -2], [12, -7, 10, -3, 1, 0, -1, 1], [5, 3, 2, 0, -2, -3, -11, 8], [2, 3, 0, -7, -6, -7, -8, 8], [18, 0, -6, -4, -1, 3, -1, -1], [-24, 19, -9, -8, -11, 4, -5, -6], [2, 11, -4, 1, -3, -2, -1, 1], [-13, 0, 10, 0, 6, -5, -1, -4], [-10, 3, -3, -6, -5, -4, -7, 2], [-6, 10, -14, -2, -6, 4, -10, -4], [21, -1, 6, 1, 2, 7, 4, 3], [-9, 15, -15, -12, -9, -3, -8, 4], [6, -1, 3, 0, 4, 0, 6, -6], [-33, 6, 0, 4, 0, -1, -3, 4], [-2, 1, 5, 0, 3, -8, -7, 0], [16, -4, -4, -2, -8, 12, -6, -2], [-4, -16, 4, 18, 3, 0, 7, -10], [-24, 0, -1, 0, -7, -1, 3, -3], [-11, 3, -2, 1, -3, 9, 4, -5], [-4, -4, 12, 4, -4, -6, 6, -6], [13, -9, 8, 4, 4, 5, -5, 0], [17, -8, -8, 4, -7, 7, -3, 2], [19, 6, -12, -10, -5, 14, -6, -3], [1, 1, 0, -5, -7, 9, -4, -7], [-23, 1, 12, 5, 10, -1, 1, -1], [11, 0, 0, 8, -4, 13, 3, -6], [-15, -2, 14, 0, 7, -5, -7, 8], [-4, 3, 9, 8, -3, -8, 3, -2], [7, -9, 2, -6, 0, -3, -1, 8], [-4, -10, 5, -3, 5, 0, 6, -3], [-8, 6, 7, -7, 0, -4, -6, 5], [-26, -11, 10, -3, 0, -3, 2, 0], [-2, 3, 7, -2, -1, 4, 7, -6], [-1, 8, -2, 6, 5, -6, 0, 9], [-6, -2, 11, 6, 1, -3, 1, -5], [-8, -5, -4, -1, 7, 0, 5, -1], [24, 11, -4, -9, -3, -6, -13, 13], [6, -4, -5, 0, -1, 1, 11, -11], [-2, -4, -4, 0, -6, 2, -10, 10], [19, 11, -1, -8, -7, -1, -2, -8], [4, 4, -2, 2, 7, 7, -11, 3], [-9, 0, -2, 4, -1, -3, 3, -2], [9, -8, 0, 0, 4, 9, -1, 8], [-9, 10, -12, -2, 1, 9, -5, -6], [1, -2, -9, 3, -8, 5, -13, -1], [4, -3, 16, 13, 5, -4, -1, 9], [1, -9, 10, 5, 6, -13, 12, 3], [0, 4, 8, 0, 8, -6, 0, 8], [18, -9, -3, 8, -6, 10, 2, -8], [11, -6, 11, 2, -1, 11, -6, 6], [-1, -7, 11, 10, 8, -5, -3, 6], [11, -12, 0, 8, -1, 9, -9, 6], [-1, 3, -11, -2, 1, 11, 6, -2], [8, 2, 0, -6, 5, -10, 11, 0], [18, 3, -1, 2, 3, 4, 9, -10], [0, -9, -1, 9, 0, -4, -6, 13], [-2, -11, 7, 14, -4, 8, -2, 0], [-25, -8, -17, 1, 0, 13, 1, -1], [-11, -8, -1, 6, -3, 4, -2, 7], [-7, 22, 1, -4, -5, -9, -2, -2], [-1, 2, -8, -8, 0, -3, 7, -2], [-1, -3, -18, 11, 3, 9, -4, -1], [-17, -12, 3, 10, 3, -2, -2, 11], [-6, -6, 4, 2, -10, 10, -6, 6], [5, -7, 3, 2, 1, 1, -10, 8], [12, -7, 3, 2, 3, -6, 15, -18], [-8, -13, -2, 13, -4, 15, 0, 0], [25, 15, -13, -10, -9, 11, -10, -6], [12, -8, 10, 6, 10, -8, 4, 10], [-8, -10, 10, 0, 1, -13, 3, -7], [16, -14, 10, 1, 3, 3, 7, 6], [-20, 14, 12, -14, 0, -10, 4, -6], [-1, 9, 11, -10, 7, -13, 4, 8], [12, -10, 19, 2, 9, -13, 15, -1], [23, 0, -7, -7, -6, 5, -3, -9], [7, -15, 8, 11, 10, 1, 4, 5], [-17, -3, -7, 6, 5, -5, 4, -4], [24, 6, -8, -6, 0, 12, 4, -4], [2, -14, 11, 1, 4, 0, 6, 5], [3, -3, 0, 9, -3, 9, -2, 1], [11, 13, 4, -13, -4, -7, 4, -3], [-28, -7, 10, 21, 7, -4, 1, 1], [0, -19, 6, 15, 12, 5, 8, 0], [6, -2, 4, 6, 8, -14, 0, 12], [8, 12, -12, 2, -8, 16, -8, -4], [5, -8, 13, 11, 12, -3, -3, -5], [8, -11, 17, 4, 11, -16, 1, 4], [-4, -22, -8, 4, -3, 12, 2, -4], [-26, -13, 10, 6, 0, -4, 8, 8], [12, -13, -1, 6, -1, 8, -11, -8], [20, -16, 19, -7, 9, -16, 0, 5], [-8, 21, -10, -19, -9, -2, -3, -5], [-21, 9, 1, -6, -1, -7, -8, 6], [18, -6, 4, 6, 10, 0, -2, 10], [14, 14, -4, 0, 3, -6, -3, 4], [6, -7, 10, 5, -2, -9, -4, 6], [15, -1, 6, -13, -1, -9, 2, 1], [-20, -16, -14, 16, 8, 12, -2, -2], [-5, -1, -1, 6, -9, 21, -14, 2], [8, -12, 28, -4, 14, -26, 18, 4], [-7, -2, 0, 12, 9, 3, -1, -2], [-4, 1, 0, -15, -5, 2, -5, 3], [33, -4, -5, -2, -1, 0, 8, -3], [21, -12, -5, 0, -5, 22, 0, 3], [1, -3, 1, 8, -3, 11, 0, -4], [12, -6, -2, 2, 8, 4, -10, 6], [-7, 12, -13, -9, -14, 9, -7, -9], [1, 4, 6, 8, 10, -3, 5, 8], [8, 0, 2, 2, 2, 0, -12, 2], [7, -5, 14, -7, 5, -1, 6, -1], [14, 14, -2, -16, 4, -14, -6, 2], [18, 11, -9, -14, -4, -6, -8, 10], [5, -18, 6, 7, 3, -1, -6, 3], [39, -3, 1, 1, 0, 0, -1, 8], [0, -11, -7, 6, -2, 12, 8, -6], [34, -5, -16, -9, 5, 10, -7, -1], [16, 12, -4, -16, 2, -10, 10, -2], [-3, -9, 2, 5, -8, -1, -13, -1], [4, 11, -9, -1, -6, 10, 10, -11], [0, -2, 0, 4, -10, 2, -12, -4], [15, 6, 10, -9, 9, -14, 14, -2], [-51, 7, -1, 6, -7, -1, -4, 2], [-19, -14, 13, 5, 5, 10, 4, -10], [17, -17, 15, 8, 5, -13, -4, 12], [-27, -4, 1, 6, -7, 2, 2, -7], [-10, 23, -14, -11, -10, 11, -8, -2], [5, 6, -3, 5, -4, 7, -3, 3], [19, -12, -1, 13, -3, 1, 1, 1], [26, -14, 4, 8, 4, 10, -10, -2], [23, 8, -20, -12, -12, 15, -11, -4], [20, 1, 9, -2, 7, -10, 7, 6], [16, -1, 19, -4, 6, -18, -10, 2], [24, -1, -7, -4, 1, -4, 13, -10], [-6, -6, -6, 8, -8, 12, -2, -10], [-26, 8, -17, -9, -13, 4, -3, -5], [-20, 7, 0, -9, -1, -2, 9, -3], [18, 9, 4, -7, 15, -14, 7, 9], [13, -4, 2, -4, 0, -1, -9, 6], [18, 19, -9, -8, 0, 0, 6, -2], [3, 8, -15, -6, -11, 4, 8, 1], [6, 7, -10, 4, -14, 12, 2, -10], [7, -21, -10, 16, -6, 23, -9, 4], [-8, -11, -1, 2, -11, 6, -1, -6], [10, -13, -21, 8, 1, 16, -9, 4], [-5, 10, -11, -12, -11, -4, 4, 1], [-25, -18, 2, 18, -1, 13, 3, -2], [-19, 0, 8, 2, -10, -7, 9, 2], [30, -16, 26, 4, 20, -24, 14, 6], [17, 9, -18, 5, 4, 13, 1, 1], [6, 24, -16, -10, -8, -8, -10, 6], [26, -7, -8, -7, -1, 18, -7, 5], [-19, -18, 5, 8, 7, -1, 18, -2], [1, -13, 4, 7, -7, 5, 0, -3], [7, -16, 7, 2, 7, 2, 2, 7], [0, 0, -4, 4, 0, 4, -4, 2], [18, -16, 8, 10, 10, 4, 2, 4], [-8, 15, -9, 0, -3, 8, -11, -6], [-2, -6, -2, 6, 7, -3, 19, -17], [12, 30, -22, -14, -11, 12, -15, -10], [1, -2, 5, -5, 3, -6, -2, 8], [36, 1, 7, 6, -5, -2, 1, 6], [23, -5, 11, -6, 7, -9, -2, 4], [-15, -30, 6, 18, 2, 7, -3, 0], [3, -18, 15, 14, 3, 7, 14, -10], [17, 18, -23, -6, -5, 4, 0, -7], [-23, -7, -6, 7, -1, 1, -2, 5], [35, -3, -5, -6, -6, 3, 3, -2], [-2, 7, 5, -4, 3, 0, -15, 0], [-12, 14, -4, 2, -1, 13, -9, -1], [4, 23, -18, -2, -8, 6, -6, -16], [13, -5, -4, -1, -9, 21, 6, -7], [-9, 15, -2, -1, -9, 7, 6, -9], [-24, -5, 23, 0, 1, -12, 11, -4], [36, -5, 4, -15, 3, -4, 1, -3], [20, -8, 15, 11, 0, -6, 10, 17], [38, -6, 10, -14, 6, -14, 0, 8], [-33, -5, 4, 11, 3, -1, -2, -7], [1, 4, 5, 8, -1, -3, 0, -12], [9, -16, 0, 0, 3, 3, 1, 2], [-17, 13, 7, -5, 8, -8, -5, -2], [8, 10, -11, -19, 6, -6, 2, -1], [25, -1, -10, 1, 9, 7, -8, 9], [47, -2, -3, 2, -3, -3, 2, 0], [7, -10, 14, -2, 15, -15, 13, 4], [6, 7, 8, -13, 1, -2, -7, 7], [-10, -13, 4, 5, -6, -3, -4, -4], [-21, -15, -16, 6, -2, 21, 5, 0], [17, 0, -24, 1, -9, 13, -16, 7], [5, -1, 0, 3, -2, 3, 8, -19], [20, -5, -8, 4, 0, 18, -14, 18], [34, 14, -20, -14, -5, 8, -21, 8], [10, -4, -16, -2, -17, 19, -11, 3], [21, -7, 2, -7, -12, 11, 2, -5], [3, 13, 3, -4, -1, 7, 0, -6], [-27, -13, 7, 8, 9, -7, 0, 8], [-4, -12, 14, 24, 12, 0, 6, -6], [-32, 10, 8, -10, 6, -8, 6, -14], [-3, -18, -3, 15, 13, 0, 4, -2], [14, 0, 24, 8, 5, -9, 3, -3], [-2, -5, 0, -6, 2, 4, 6, 4], [-8, -11, 4, -3, -1, -4, 13, -3], [16, -16, 10, 15, 5, 15, -3, -4], [-15, -2, -15, 14, 9, 3, -2, -2], [42, -14, 9, 17, 6, -4, 8, 9], [18, 19, -14, -7, -7, 0, -7, -5], [20, -17, 2, -1, 1, -8, 15, -9], [12, -6, 9, -3, -10, 10, -12, 5], [20, 24, -22, -16, -6, 10, 4, -18], [18, -12, 6, 8, -2, -4, 0, 8], [22, -10, 16, -6, 2, -6, 0, 16], [5, 5, 0, 7, 1, -7, 2, -3], [-6, 6, -10, 16, 3, 12, -5, -2], [-19, -3, -5, 3, -8, 10, 1, -6], [-19, -2, 14, 4, -3, 11, -9, 6], [1, 11, 0, 6, 6, 3, -1, 0], [11, -33, 1, 4, 2, 5, 5, 2], [23, -30, 23, 17, 21, -14, 2, 12], [19, -9, 4, 13, 1, 3, -6, 11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6004_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6004_2_a_d();".
// 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_6004_2_a_d(:prec:=8)
    chi := MakeCharacter_6004_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_6004_2_a_d();".
// 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_6004_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6004_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<5,R![248, 340, -241, -248, 87, 56, -15, -4, 1]>],Snew);
    return Vf;
end function;