
// Make newform 8004.2.a.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8004_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_8004_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_8004_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 := [-4, 18, -10, -35, 19, 19, -8, -3, 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], [50, -108, -131, 131, 79, -46, -13, 5], [-56, 106, 133, -131, -79, 46, 13, -5], [50, -102, -122, 126, 76, -45, -13, 5], [-104, 210, 273, -269, -171, 98, 29, -11], [-132, 272, 317, -339, -195, 120, 33, -13], [71, -143, -175, 179, 107, -64, -18, 7]];
    Rf_basisdens := [1, 1, 2, 2, 1, 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_8004_a();"
function MakeCharacter_8004_a()
    N := 8004;
    order := 1;
    char_gens := [4003, 2669, 3133, 553];
    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_8004_a_Hecke(Kf)
    return MakeCharacter_8004_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], [1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, -1, 0, 0, 0, 1], [-1, 0, 0, 0, 1, 0, 0, 0], [-1, 2, 0, 1, -1, 0, -1, 0], [-1, -1, 0, 1, 1, -1, -1, -1], [1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 1, -2, -1, -1, -1], [-2, 0, -1, -1, 0, 2, 1, 1], [-1, -4, 0, 1, 0, 1, 0, -1], [-1, -1, 0, 0, -2, 2, -1, -1], [-2, 0, 2, 0, -1, 2, -3, -1], [-1, -4, -1, 1, 1, -1, 1, 0], [2, 0, 0, -3, 3, -2, 4, 2], [0, -1, 1, -2, 2, 0, -1, -1], [-1, -2, 0, -2, 2, 2, 3, 0], [0, 3, 1, 3, -3, 0, -3, 1], [-5, 5, -4, -1, 0, 0, 1, 3], [-2, 4, -1, 0, -3, 1, -3, 1], [2, 0, -2, -3, 3, -2, 7, 4], [-2, 1, 1, 6, -2, -2, -1, 0], [0, 0, 3, 2, -2, -1, -4, -3], [0, -2, -3, -4, 5, -3, 7, 4], [1, 3, 2, 1, -3, -1, -4, 1], [-4, 1, 2, 4, 0, 0, -6, -6], [-2, 1, 1, -2, 0, 0, 0, 0], [-2, 0, -1, 0, 2, -2, 0, 2], [-3, 1, 4, 1, 2, -2, 3, -1], [-3, 1, 2, 0, 0, 3, 0, 0], [-3, 1, 1, -2, 1, 0, 0, -2], [-8, 4, -1, -1, -1, 1, 0, -2], [-2, -2, 4, 0, -1, 6, -2, 0], [-4, -4, 0, 2, 0, -2, -1, -4], [1, -4, 1, -3, 3, -3, 4, 4], [-9, 0, -4, 4, 3, -3, -1, -3], [-3, 7, -1, 1, -2, -1, -4, 4], [1, -7, 3, 0, -2, 2, 0, 0], [-7, -4, -3, -4, 2, 3, 1, 0], [-2, 2, -3, 0, -3, -2, -4, 2], [1, 1, -4, -2, 4, -5, 4, 1], [2, 0, 7, 1, -6, 4, -2, -1], [-6, 5, -7, 2, 2, -3, 1, 3], [-7, 1, -4, 2, 0, 2, -1, -4], [-5, 1, 2, -4, -4, 3, 3, -1], [-8, 3, 0, -4, -1, 0, -4, -1], [2, 0, 4, 3, -7, 3, -6, -6], [-2, -3, 4, -3, 0, 4, 2, -2], [3, -1, 2, -1, -1, -2, 4, -3], [-3, -6, 4, 5, 0, 3, -1, -3], [-4, -1, -6, -6, 5, -2, 5, 5], [3, -8, 5, 6, 0, 6, -1, -5], [-7, 8, -5, -5, 1, 0, 5, 8], [5, 1, -1, -3, 1, -8, 4, 2], [-2, 1, -1, -2, 0, -1, 9, 8], [-3, -7, 2, 1, 4, 5, 0, -3], [1, -6, 2, -3, 5, -3, 2, 2], [-6, 0, -10, -5, 4, -1, 6, 6], [-7, -8, 4, 4, 1, 4, -2, -7], [8, 1, 7, -3, -4, 1, -7, 0], [-4, -3, 1, 3, 3, -10, 0, -7], [3, -1, 4, -5, -2, 6, 0, 4], [-2, -3, 1, 5, -1, 1, 2, 0], [-2, -5, 1, 7, 2, 1, 0, -8], [-10, 3, 0, 4, 4, -3, 3, -2], [-2, -3, 4, 4, 8, -5, 2, -6], [-5, 8, -7, 0, 0, -5, -3, 2], [-2, 5, 2, 8, -13, 3, -10, -6], [3, -2, 5, 2, -7, 4, -1, -6], [8, -1, 1, -6, -2, 1, 7, 3], [-7, 12, -10, 0, 1, -1, 7, 7], [5, -1, 5, 2, -10, 2, -3, -4], [-6, -4, -1, -3, 9, 0, 7, 1], [-2, -6, -5, 2, 1, -1, -2, -7], [-15, 9, -1, 7, -4, 6, -10, -1], [1, -6, 4, 2, 1, 5, -3, -1], [-1, 14, -3, 0, -1, -8, -1, 6], [8, 8, -3, -2, -3, -6, 5, 13], [4, 1, 4, -2, -11, 7, -3, 2], [-13, 0, 4, 0, 0, 1, -10, -9], [7, -2, 6, 1, -2, 4, -3, -9], [3, 0, 0, -1, -4, 2, -3, -1], [8, -8, 6, 5, -2, 2, 3, -7], [3, -15, 3, 2, 4, 5, 1, -3], [-6, -1, -8, -10, 2, 2, 7, 9], [-10, 5, -5, 1, 2, 1, 10, 0], [-5, 6, -5, -7, -1, 1, -7, -4], [-6, 6, -2, -2, 3, -3, 5, 5], [-8, -6, -5, 0, 5, -2, -4, -3], [4, 15, -11, -1, -4, -1, 7, 11], [-8, 9, -3, 4, 2, -4, -1, -5], [1, -4, 2, 2, 7, -2, 2, -9], [0, -5, 9, -1, -7, 8, 0, -6], [1, 5, -13, -3, 9, -10, 12, 7], [-5, 0, -6, -12, 5, -4, 7, 11], [5, -1, 2, 0, -1, 5, 7, -3], [10, -3, -2, 2, -1, -5, -6, -3], [7, 9, 0, 8, -2, -8, -4, -3], [0, 2, -5, -6, -5, 0, 5, 6], [1, -8, 11, 3, -2, 0, -2, -12], [7, 0, 5, -6, 1, 0, 5, 3], [-8, 2, -2, 6, 1, -7, -2, -5], [-10, 1, -1, 0, 5, 5, 2, 5], [4, 0, -6, -5, -1, 4, -1, 4], [1, 1, 7, 1, 2, -5, -4, -2], [1, 6, -8, -3, 2, 3, 5, 5], [2, 12, -5, -1, 1, -2, 3, 12], [-14, -7, 2, 3, 3, 6, -1, -2], [-2, 6, -10, 0, -3, -1, 3, 0], [4, 0, -8, -3, 6, -10, -2, -3], [0, -6, -5, -8, 11, -5, 8, 11], [10, 12, 8, -6, -2, -6, 3, 12], [-7, -1, -3, 1, 3, 3, -2, -1], [5, 2, 0, 7, -4, -4, -5, -2], [1, 0, 1, -8, -9, 9, -1, 1], [5, -6, 2, 2, -5, 6, -9, -5], [-4, 6, 2, 2, -5, 4, -9, -13], [8, 6, 7, 8, -10, 3, -14, -10], [14, -17, 6, 3, 4, 0, -2, -5], [-5, 5, -2, -8, 2, 1, 1, 10], [4, -8, 11, 3, -8, 15, -6, -8], [0, 4, 5, -2, -5, -3, -10, -5], [-5, 0, -4, -2, 2, -6, 1, 0], [-11, 10, -3, 0, 5, -1, -2, 2], [-1, 10, -5, -4, 2, -3, 10, 18], [12, 9, -3, 7, 2, -4, -5, 3], [-2, -1, -2, -4, 1, 9, 9, 8], [5, -6, 15, -1, -5, 3, -1, -8], [-2, -3, 5, 1, -5, 3, -5, -1], [2, -11, 0, -5, 12, 0, 5, 1], [1, -11, -1, 13, 2, 8, -7, -12], [14, -7, 5, -4, 5, 10, 10, 11], [11, -3, 3, -9, 1, 11, 8, 10], [-10, -3, 4, 5, 7, 2, 4, -6], [23, -1, -9, 3, -3, -8, 2, 6], [4, -4, -3, 0, 0, -1, 12, 9], [14, -8, 2, -1, 4, -5, 8, -6], [0, -13, 18, 3, -5, 5, -9, -13], [2, 6, -6, 0, -1, 1, -12, 5], [1, -13, -2, 3, 8, -2, -2, -15], [14, 0, 1, -3, 1, -1, 15, 17], [14, 4, 6, 6, -11, -1, -18, -14], [-3, 13, 0, -2, 2, -2, -2, -3], [1, 11, -13, -5, 3, 5, 6, 7], [3, 0, -14, -10, 6, -7, 11, 11], [2, -19, -6, -5, 12, 2, 13, 8], [-7, -3, -3, -2, 5, -2, 13, 2], [-7, 3, 1, 11, -7, 10, -10, -4], [-5, 7, 2, 3, -12, -5, -21, -5], [0, 0, 7, 3, -5, 12, -4, -16], [-8, -22, 2, -3, 13, 5, 5, 0], [7, -1, 8, 0, -6, 6, -3, -2], [-6, -5, 0, 2, -10, 1, 0, 6], [1, 4, -13, -6, 1, -6, 11, 5], [10, -3, -1, -11, 6, -6, 15, 6], [-18, 1, -1, 9, -8, 6, -9, -11], [6, -1, -7, 5, 3, -16, 3, 2], [14, 21, 3, 1, -10, -1, -5, 11], [8, 7, -3, 12, -8, -7, -9, 4], [5, 16, 1, -3, -5, -4, -12, 6], [-1, 8, -3, -1, -1, -8, -6, 6], [-6, 9, -8, 4, 7, -5, 5, 6], [6, -5, -3, 7, -6, 0, -6, -7], [24, 5, 1, -5, 1, 0, 9, 5], [-2, 3, -7, 6, 0, -21, -5, -6], [-6, -2, 19, 6, -5, 9, -8, -5], [-8, 10, -2, 5, 0, -2, -2, 1], [14, -6, 2, 7, -2, -5, 1, -5], [0, -11, 4, 13, 7, 1, -12, -20], [7, 2, 9, 6, -2, -7, 4, -1], [18, -9, -4, -2, 5, -2, 10, 10], [-8, 5, -9, -1, 4, 5, 4, 9], [5, -9, 4, 9, 3, -7, 6, -5], [10, -9, 6, -4, -5, -5, -1, 0], [-1, 9, 1, -7, 2, -3, 1, 7], [-2, -9, 6, -4, -8, 7, -2, -13], [11, -8, 1, -4, 2, 14, 13, 12], [17, 1, -3, -3, -7, -7, -2, 3], [13, 0, 5, 4, 2, 4, -5, -4], [-2, 2, -6, -12, -5, 8, -6, -1], [-4, 4, -3, -2, 3, -2, -4, 0], [-11, 2, 8, -3, -7, 7, 1, -11], [25, -3, 12, 4, 4, 1, -7, -7], [1, 3, 0, -12, 5, -4, -2, 11], [-11, -4, -1, 3, -9, 4, -8, -4], [-2, -4, -2, 4, 12, -4, 2, 10], [1, 7, 13, 5, -19, 6, -16, -16], [9, 1, 12, 15, -14, 9, -24, -13], [-15, 4, 4, -7, 8, -5, 4, 2], [-5, -6, -16, -8, 10, 9, 8, 6], [14, -6, 1, -6, 1, -13, 14, 9], [-18, 1, -6, -3, 13, -5, 11, 3], [-2, -1, 3, 7, 1, 5, -9, -8], [13, 1, -3, -17, -3, 11, 9, 12], [-1, 2, 1, 14, -1, -16, -3, -4], [-11, 12, -5, -5, -3, 8, -3, -3], [7, 8, 1, -2, 1, -14, 1, 2], [0, -9, 6, -11, 7, 9, 9, 0], [-13, 4, -5, 5, 6, -6, 1, 6], [-10, -10, -9, -2, 11, -2, 17, 8], [18, -6, -8, -16, 14, -9, 27, 16], [-11, 8, 1, -8, 1, 2, 12, 24], [13, -14, 10, 8, 0, 11, 5, -10], [-18, 6, -6, 7, 2, -2, -12, -10], [-3, 8, -5, -6, 10, 0, 0, 4], [27, 4, 4, -5, -2, -7, -11, 7], [6, -18, 18, -1, -8, 2, -6, -8], [-16, 11, -2, 3, 3, -3, 10, 18], [11, -5, 2, -7, 2, 0, 13, 15], [1, -3, 9, -7, 4, -1, 4, 10], [9, 0, -3, -3, 17, -17, 8, 2], [-5, -2, 15, 3, 0, 7, -13, -13], [0, 12, -8, 2, -10, 6, -1, 3], [-11, -3, -12, -1, 11, -11, 5, 8], [-7, 2, -12, -3, 0, -3, 4, 0], [-1, 5, -10, -13, 7, -7, 17, 14], [21, 11, -6, 7, 2, -8, -1, 17], [-2, 9, 1, 7, -12, 1, -16, -4], [-9, -3, 0, 0, -4, 8, -18, -13], [2, -3, -3, 9, 7, -2, 1, 4], [8, -11, -17, -5, 20, 2, 13, 10], [10, -4, 12, 8, -4, -2, -9, -2], [-12, 0, 2, 6, 0, 5, 4, 4], [-34, 6, -2, 7, 0, 4, -5, -2], [-26, 7, 8, 2, -7, 0, -5, -16], [-15, 6, 1, 6, -4, 1, 2, 0], [22, -26, 4, -10, 11, 4, 17, -4], [5, 2, 4, 6, 3, 1, 1, -11], [0, 5, -18, -8, 10, -16, 14, 4], [19, -17, 6, -12, -3, 12, 8, 7], [26, -7, 10, -1, -4, -2, 6, -6], [-22, 8, -4, 8, 3, -7, 7, 2], [16, -7, 6, 3, 6, -10, -10, -11], [10, 1, -6, -1, -3, 3, -7, -1], [5, 11, 1, -5, -13, 2, -13, 12], [21, -5, 6, -2, 1, -7, 0, 7], [6, 1, -5, 2, -7, -7, -21, 5], [8, -23, 7, -7, 3, 4, 5, -12], [9, 1, -1, 7, 3, -4, 2, -5], [20, -7, 11, -9, -12, 7, -4, 1], [-24, 2, 0, 2, 1, 7, -25, -17], [21, -21, 6, -1, 0, -8, 15, -1], [14, -16, 1, -5, -3, 4, 4, 3], [23, -13, 11, -15, 2, 9, 8, 6], [13, -20, 5, -13, 18, -3, 9, -4], [-2, -5, 7, -8, -9, 7, -12, -9], [19, 3, 1, -1, 4, -6, 7, 9], [17, -10, -9, -8, 8, -4, 25, 11], [-9, 25, -11, 3, -4, -2, 8, 18], [-26, 8, -6, 1, -8, 8, -11, 4], [22, -8, 12, -17, 4, 3, 2, 11], [-7, 9, -4, 2, 9, -12, -2, -7], [-8, 6, -1, 6, -18, 0, -21, -23], [-2, 2, -21, -5, 11, -4, 12, 11], [0, 11, 3, 0, 1, 6, -5, -11], [4, -1, 3, 5, -12, 11, -17, -1], [-4, -10, -5, -7, 21, -5, 10, 7], [0, 7, 1, 10, -9, -9, -18, -10], [5, -15, 9, 3, -3, 2, -14, -4], [0, 10, 2, 14, -12, -2, -6, -19], [1, -9, 12, 10, 1, -5, -1, -4], [-8, -3, 0, 8, -7, 3, -9, -32], [12, -21, 3, -3, 10, -2, 4, -9], [-18, 20, -12, 6, 0, -16, -3, 6], [-10, 2, -1, -1, -3, -8, 6, -12], [-10, 9, -11, -7, -1, 0, 9, 9], [-14, 14, 6, 20, -25, 0, -24, -10], [-4, 17, -7, 16, -17, 0, -11, -1], [-3, -7, -17, 10, 5, 4, 5, -5], [-14, 3, 7, 13, -9, 8, -33, -22], [-8, 1, 5, 7, -2, -1, -3, 1], [18, -11, 7, 5, -3, 7, -14, -13], [-3, 4, -5, 9, -4, -3, -5, 0], [18, -16, -4, -9, 15, -5, 12, -10], [-21, -7, -5, -1, 17, -12, 22, 4], [-4, 0, 20, 2, -13, 17, -6, -7], [11, -6, 6, 14, -11, -3, 5, -5], [9, 22, -17, -3, 3, -16, 4, 16], [31, -12, 5, 12, -6, 0, -19, -12], [18, -17, 1, 8, 4, -2, -8, -4], [-18, 28, -5, 3, 1, 0, -2, 1], [4, -1, 1, 25, -8, -1, -2, -2], [-8, 8, 4, -14, -5, 16, 10, 4], [8, -32, 16, -1, 8, 11, 3, -23], [17, 11, -1, -15, -1, -2, -3, 14], [-24, -9, 3, 13, 5, 1, -19, -20], [10, 5, 8, -3, -11, 7, -9, 9], [1, 3, 10, 13, -3, -4, -4, 1], [-4, 13, -2, -3, 7, -1, 2, 18], [28, -15, 3, -19, 5, -4, 26, 9], [10, 1, 7, -4, 3, 11, -3, 17], [21, -3, -10, -10, -2, 9, -4, 7], [14, -2, 13, 5, -2, -2, -6, -13], [-21, 14, 6, -9, 11, -4, 15, 5], [-2, -7, 8, 5, 1, -14, 8, -10], [0, 17, 9, 7, -19, 6, -27, -11], [6, -3, -16, -14, 23, -21, 27, 18], [-13, 4, -7, -23, 4, 1, 1, -4], [24, 26, -9, -1, -16, -2, -5, 15], [17, -11, 6, 1, -7, 4, -9, 1], [4, 3, 3, -5, -6, 23, -3, -3], [-9, 4, -10, 7, -3, 10, -3, -11], [16, 0, 6, -5, 17, -11, 8, 8], [7, 10, 17, 20, -11, 8, -28, -14], [9, 3, 5, -13, -5, 8, 3, 1], [-3, -6, -16, 4, 4, -8, 4, 9], [32, 11, 4, 1, -5, -4, 9, 18], [31, -6, -4, -21, 10, 5, 14, 4], [10, 0, -8, 14, 0, -10, -6, 8], [-6, 13, -12, -7, 12, -19, 15, 7], [-3, -6, 1, 8, 4, -4, 1, -22], [-20, 1, -2, -13, 1, 7, 8, -1], [11, -21, -11, -20, 18, 2, 17, -4], [13, 2, 8, 0, -7, 6, -5, -2], [25, -20, 4, 2, -3, 2, -2, -19], [30, 31, -4, 1, 0, -18, 3, 8], [23, -6, 16, -7, 0, 7, -3, -4], [-15, 6, -4, 2, -7, -5, -4, 12], [13, -10, -2, -8, 8, -9, -7, -1], [-7, 10, 8, 3, -16, 11, 11, 4], [-6, -17, 16, 15, -11, -9, -9, -18], [-4, 29, -15, 7, 10, -20, 2, 7], [11, -16, 4, -9, 5, -1, 12, 11], [-17, -3, 2, 22, -4, 0, -20, -17], [24, 4, 16, 14, -5, 6, -24, -22], [8, -13, 15, 22, 4, -2, -12, -19], [17, 13, -21, -2, 9, -8, 12, 17], [-25, 11, -7, 8, -8, 15, -14, -6], [-12, -4, -22, -20, 27, -6, 34, 24], [21, 13, -3, 2, -3, -2, 8, 24], [-28, 10, -13, 12, -18, 5, -13, -17], [-9, 7, 9, 12, -23, 9, -1, -7], [5, 0, 2, 13, -20, 2, -6, -9], [-10, -1, 2, 22, -3, -17, -15, -18], [-1, -21, 16, 7, 2, -1, -9, -26], [0, -20, -4, 13, 5, -4, -18, -23], [-6, -12, -1, -17, 8, 5, 1, -13], [34, 4, 2, -9, 11, -5, 15, 5], [23, -17, 18, 6, -1, -7, 5, -9], [-40, 20, -7, 9, 7, -6, 4, 3], [-4, -6, 5, 15, -7, -3, -4, 4], [-15, 9, -10, 7, -12, 8, -20, -4], [22, -5, 3, 14, -6, 9, -2, 3], [-7, 1, -10, -10, 11, -16, 18, 33], [10, 13, -13, 4, 3, -6, 7, 25], [1, -10, -15, -8, -4, 5, 20, 14], [24, 7, 7, -1, -1, -16, 13, 2], [-5, 0, 7, -6, 20, 0, 11, 11], [19, -24, 3, -17, 30, -7, 16, -4], [12, -3, 18, -2, 2, 12, -2, 17], [6, -6, 5, -5, 28, -10, 8, 1], [-8, 8, -24, 7, -2, -20, -3, 8], [-9, 8, -22, -5, 17, -12, 15, 11], [-5, -5, -14, -3, 24, -18, 23, 11], [-11, 22, -13, -2, -7, 12, 9, 15], [29, -2, -10, -8, -8, 5, 9, 21], [8, -16, 4, 6, -3, -8, 7, 3], [7, 6, 11, 3, -17, -6, -3, -16], [-2, 27, 8, 4, -16, -9, -14, -9], [-43, 11, -4, 2, -9, 6, -6, -9], [-1, 10, -8, -28, 12, 11, 16, 14], [12, 18, 8, -4, -13, 1, -19, 0], [-12, -1, 2, -15, 10, 2, 11, 8], [-29, 9, -4, 11, -9, -1, -34, -13], [-20, -13, -8, -2, 19, 4, 27, 3], [6, -7, -12, -13, 27, -2, 26, 29], [29, 6, 5, -19, -4, -3, -2, 6], [-18, 0, 11, 21, -4, -1, -13, -3], [10, 7, -6, -6, 7, -2, 18, 13], [-2, 21, -14, -14, 11, 4, 7, 12], [0, 8, -8, -1, 0, -23, 4, 19], [3, -10, -9, -9, 24, -10, 5, 17], [5, 7, 16, -4, 2, 6, -13, -5], [-6, 16, 13, 26, -25, 7, -41, -14], [8, -35, 10, 2, 2, 0, 3, -15], [12, 16, -2, -9, -3, -1, 3, 26], [3, 4, -1, -6, -14, 14, -7, 12], [3, 0, -22, -4, 16, -13, 10, 21], [19, 4, 11, 19, 0, -20, 1, -10], [6, 2, -20, -15, 32, -20, 27, 18], [-1, 9, 28, 16, -20, 13, -17, -15], [-17, 15, -22, -13, 9, -7, 11, 18], [-21, 4, 5, -6, 15, -3, 4, 8], [7, 25, -4, 15, -15, -2, 6, 7], [-10, 40, 4, 7, -7, -6, -17, 0], [-31, 17, 5, 8, 1, 12, -18, 9], [7, -8, 0, -5, -10, 18, -2, 15], [-4, -1, 21, 11, -8, 1, -17, -14], [10, -8, -1, -9, -10, 10, 21, 29], [-4, -4, -20, -8, 27, -16, 32, 30], [-19, -11, -2, 7, 10, 8, 2, -20], [-32, 14, -6, 4, -4, 5, -31, 2], [9, 11, 1, 15, -3, -12, -20, 5], [-13, 15, -17, 10, 10, -12, 12, 5], [-14, -5, -10, -4, 13, 9, 11, 0], [-3, 26, -2, 6, -3, 2, -13, 1], [5, 11, 2, -5, -24, 15, -13, 8], [-15, -4, 10, -12, 0, 7, -24, -6], [29, -10, 14, 4, -9, 1, -29, -23], [-3, 21, -6, 10, -15, 7, 5, 7], [-26, -20, -3, -10, 16, 17, 17, 1], [-31, -28, -7, 4, 9, 9, 14, -10], [25, 5, 7, -7, 7, -10, 14, 26], [-16, 16, -19, -10, -11, 9, 11, 15], [-8, -9, -4, -6, 7, -23, 10, -9], [23, -25, 16, 3, 3, 18, 14, -16], [-15, -5, -32, -9, 12, -3, 15, 13], [-39, -6, -3, -20, 8, 21, 1, 14], [-24, -23, -1, -1, 2, -6, 12, -2], [-19, 8, 1, -1, 5, -11, -7, 4], [31, -24, -2, -18, 10, -9, 35, 7], [24, -6, 4, -5, -4, -1, 14, 20], [-21, -3, -4, -4, 9, -14, 15, 3], [-16, 18, -9, -14, 15, 8, 2, 18], [6, -11, -14, -20, 19, -22, 21, 25], [-24, 0, 4, 15, -15, 12, 9, -3], [-12, 5, -20, -17, 6, 10, -1, -3], [15, 32, -6, -14, -1, -9, 15, 13], [6, -34, 18, 4, -14, 10, 0, -16], [3, -17, 3, 15, -11, 14, -3, -14], [27, 17, 4, -7, -23, 9, -14, -3], [-10, 33, -13, 7, 10, -17, -5, -1], [31, -45, 10, -9, 1, 14, 11, -14], [-2, -2, 5, 3, -13, 10, -6, 9], [-26, 6, -1, -6, -4, 5, -16, -5], [-24, 7, -15, 8, -5, -5, 3, -6], [-16, 9, 16, -3, 7, 7, -1, 7], [2, -1, 14, 18, -23, -7, -25, -28]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8004_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8004_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_8004_2_a_d(:prec:=8)
    chi := MakeCharacter_8004_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_8004_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_8004_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8004_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-4, 10, 122, 107, -35, -52, -5, 5, 1]>,<7,R![-6, 54, -145, 104, 56, -41, -11, 4, 1]>],Snew);
    return Vf;
end function;