
// Make newform 8004.2.a.e in Magma, downloaded from the LMFDB on 19 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_e();".
// 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_e();".
// 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 := [-11, 9, 58, -39, -79, 40, 32, -13, -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], [16720, -13414, -21492, 13848, 8484, -1742, -1649, -7, 93], [-29868, 29593, 118464, -26245, -80862, 5373, 14451, -54, -676], [-4120, -41031, 26624, 93731, -37863, -42227, 15049, 4027, -1248], [16986, 53254, -66924, -97748, 67542, 45249, -21416, -4474, 1613], [19075, 34963, -83539, -83900, 76026, 43507, -23065, -4481, 1706], [-12313, -18907, 68801, 92100, -79649, -53822, 27005, 5663, -2082], [35858, 64804, -122465, -136012, 102091, 60980, -29524, -5931, 2159], [24219, 58502, -107727, -144212, 105714, 71295, -33464, -7113, 2535]];
    Rf_basisdens := [1, 4877, 4877, 4877, 4877, 4877, 4877, 4877, 4877];
    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, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 1, 1, 0, 1], [-1, -1, 0, 0, 1, -1, 0, -1, 0], [0, 0, 0, 1, 0, 1, 0, 1, 0], [0, 1, 0, -1, -1, 0, 0, -1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 2, -1, -2, 0, -1], [-1, -1, 1, 0, -1, 0, 1, 0, 0], [-2, 0, 1, -2, 0, -1, -1, 0, -1], [-1, 1, 1, 1, -2, -3, 0, 1, 0], [0, 1, -2, -1, 1, 0, -1, 0, -1], [-3, 0, 0, 0, -1, -1, -1, 0, 2], [-1, 0, 0, 1, -1, 0, 1, 1, -1], [-1, -1, 0, 1, -1, 0, 0, -1, 1], [2, 1, -2, 1, 0, 3, 1, -1, -1], [-1, -1, 0, 0, 0, 1, 0, -1, -1], [0, -1, -1, -1, -1, -1, -2, 1, 0], [-2, -1, 2, 2, 0, -1, -1, 0, -2], [-1, -1, 2, 4, -1, -2, -1, 1, 1], [0, -3, -2, 0, 2, 1, 3, -1, 2], [0, -2, 2, -2, 2, 3, 0, -3, -1], [-6, 1, 0, 3, -3, 1, 1, 3, 2], [0, -3, 0, 1, 0, 1, 1, -1, 1], [0, 0, -1, 1, 3, 1, 2, 0, 0], [-2, -1, 2, 0, 1, 2, -2, -1, -1], [1, -1, -3, -1, 4, 0, 1, -2, -2], [2, -1, 1, -1, -1, 1, -2, -1, 0], [1, 2, 0, 2, 1, -2, -1, 1, -1], [-2, -1, 0, -1, -5, -1, 0, 1, 3], [0, -1, -2, -1, 1, 1, 1, -2, 2], [-4, -3, 0, 2, 3, -3, 0, 1, 0], [2, 1, -2, -5, 0, -2, 0, 2, 0], [0, 4, 2, 2, -2, -1, 1, 3, 0], [-3, -1, -1, -2, 3, -3, 0, -1, 0], [2, 2, 1, -4, 1, 0, 2, 0, -1], [-5, -3, 0, -2, 1, 0, -2, 1, -1], [0, 2, -2, 1, 2, -1, 0, 3, -1], [-1, 2, 1, 1, -1, -1, 5, -1, 0], [3, 2, -2, -3, -2, 0, -1, 2, -1], [0, -3, 1, 0, 8, 0, -3, -1, -2], [-2, 3, 3, 2, -5, 2, 0, 1, -1], [3, 1, 3, 6, -3, 3, 2, 4, 2], [-4, -4, 2, -1, 3, -2, -3, -5, 1], [0, 0, 3, 5, -2, 1, -1, 2, 1], [-2, -4, 3, 0, 3, 0, -1, -5, 1], [3, 2, -1, -3, -4, 3, 0, 2, -2], [-2, -1, 1, 0, 4, -2, 0, 0, -4], [-11, -1, 1, 1, 0, -2, -6, 1, 0], [4, 1, -1, 2, -6, 3, 5, 2, 1], [1, 2, -3, -1, 3, -3, -4, 0, 1], [5, 2, -3, -3, 1, 2, 4, -1, -1], [-3, 3, -2, 1, 6, 1, 2, 1, 1], [-2, 0, -3, 0, 5, 3, -1, -2, -4], [4, -2, -2, 1, 3, 2, 5, 2, 1], [-6, 0, -3, 0, -1, 5, 0, 0, 0], [-8, 3, -3, 0, 2, -3, 3, 0, -1], [1, 2, 0, 0, -1, 6, 1, 2, -1], [-2, 6, -5, 0, -6, 4, 6, 3, 1], [-3, 0, 0, 3, -6, 3, 0, 1, 1], [-7, 1, -1, -8, -5, 1, 2, -3, 0], [-2, -3, 3, 2, 0, -2, 1, -2, 4], [-7, 4, -3, 0, 3, -2, 0, 0, -3], [-3, -1, 6, 3, 1, -3, 0, 0, 2], [-2, 5, 2, -3, -6, 3, 0, -3, 0], [-6, 0, -2, -5, -4, 1, -4, 3, -3], [0, 5, 0, -4, -2, 3, -1, 0, -1], [-6, -5, 1, 10, 1, 3, 1, 5, 4], [0, 2, -5, -1, -4, 1, 5, 5, 2], [5, 5, 1, -4, -7, 3, -2, 5, -2], [-8, -4, 0, 1, 5, 1, 3, -4, 2], [-8, -4, 7, 0, 2, -4, -4, -4, 1], [-2, 5, 0, -2, -3, 0, 0, -2, -1], [2, 4, -4, -4, -2, -7, -5, 2, 0], [-4, 5, -1, 6, -3, -2, -3, 4, 1], [-17, 1, 5, -4, 1, -2, -4, 0, -1], [-4, 2, -5, 4, 3, -1, -1, 3, -2], [-7, 0, 6, 1, -2, -2, -2, 1, 4], [-9, 0, -1, 6, -1, 5, 1, 4, -1], [-3, 7, -2, -4, 0, 7, 3, 0, 1], [-6, 5, 4, 0, -6, -2, -4, -1, 0], [-5, -7, 6, 2, -4, 0, 2, 1, 2], [2, -1, 3, -4, -3, -1, -3, 4, -6], [-8, 0, 1, -4, 7, 2, -3, -7, -3], [2, 8, -5, -4, -3, 3, -1, -1, 0], [-9, -4, 8, 4, 3, -1, -5, 0, -2], [9, -3, 1, -2, 3, 2, -7, -2, -3], [3, -4, 1, 0, 2, 3, 7, 0, 2], [-8, 0, -1, 7, -1, 6, 2, 6, 1], [14, -1, -2, 9, 5, -1, 1, 4, 0], [-10, -2, 2, 2, 5, -3, 0, -1, -2], [0, -3, 8, 3, 0, -9, -8, -1, -3], [-6, -7, 0, 0, 1, 0, 7, -7, 6], [-13, -5, 0, -1, 1, -6, -5, -4, 5], [-11, -9, 0, -3, 0, -2, 0, 3, -4], [0, 2, -2, 0, -4, 4, -4, 1, 2], [-8, -1, -3, -1, -1, -5, -4, 6, -2], [-3, 1, -1, 0, 3, -1, 1, -1, 4], [-8, 2, -1, 0, -8, 6, 7, 1, 4], [-1, 3, -5, -3, 10, -2, 2, 1, -5], [-6, -4, 1, -1, 5, -9, -4, 1, -2], [-15, -5, -5, 1, 8, -4, -3, -7, 2], [-6, -5, 0, -2, 5, -7, -8, -2, 2], [-4, 4, -1, 2, -5, 2, 2, 4, -1], [-10, 1, 3, -3, 0, -5, -3, 0, 5], [10, 6, 1, -1, -3, -4, 2, 7, -3], [5, 1, 1, -5, 4, 3, 5, -11, -3], [0, -6, -1, 8, 5, -2, 3, 0, 1], [-1, -4, -8, -5, 16, 2, 2, -3, -2], [-1, 5, -4, 2, 5, -1, 3, -1, 2], [3, 0, 4, -2, -1, -2, 2, 1, 6], [1, 4, -3, 0, -6, -3, -4, 3, 1], [-13, -7, 2, -1, 3, -1, -1, -5, -1], [-4, 2, 4, 7, -5, 3, 1, 7, -4], [3, -7, 0, -1, 1, 0, 5, -4, 5], [-14, -1, -2, 1, 5, 2, -5, 0, -1], [-1, -2, -2, -3, -3, 6, -1, 9, -2], [-8, -8, 9, 12, 3, 2, -3, -6, 1], [2, 3, 0, 3, 1, 3, 0, -4, -4], [-2, 2, -4, 1, -6, -2, 1, 3, -2], [-2, 7, 0, 0, -11, 0, -4, 8, 3], [3, 7, -6, -1, 2, -4, 4, 2, -6], [-6, 0, -2, 2, 1, 12, 6, -4, 3], [-5, 5, 6, 2, -8, 4, 5, -4, 2], [2, -2, -2, -6, -1, -7, -7, 1, 0], [8, 2, 2, 3, -8, 3, 1, -2, 1], [-5, 6, -2, 2, -13, -1, 7, 2, 8], [11, 1, -3, -5, -2, 2, -3, 2, -2], [0, 0, 5, -2, 0, -1, -2, -2, -8], [6, -2, 4, 7, -2, -2, 2, -7, 3], [4, 8, 0, -5, 1, 3, 4, 1, -1], [-16, -9, -3, -2, 4, -5, -4, -1, -3], [-12, 7, 5, 3, -5, -7, -1, 2, -3], [7, -7, -4, 1, 6, 9, 4, -5, -1], [-2, -4, -6, 6, 10, -5, 4, -1, 1], [9, 3, -1, -2, 0, -6, -4, 3, 4], [10, 4, 4, 4, -10, 8, 8, 0, 2], [6, 0, 3, -7, 3, 3, -4, -3, -4], [-5, -2, -7, -6, 1, 4, 5, -4, -1], [2, 7, -2, 10, 3, 6, 8, 3, 3], [-4, 2, -5, -2, -4, -1, 0, 10, 4], [2, 5, -5, -5, 5, -5, 8, 6, -2], [3, -4, 1, -2, 8, -1, -8, 7, -5], [11, -2, -3, 2, 12, -6, -8, 2, -6], [-7, 0, 5, -5, -1, 6, 1, -2, -1], [-10, 5, -4, 1, -2, -6, 5, -1, -2], [-12, -5, 2, 7, -2, -10, 1, -2, 4], [1, -5, 10, 6, -13, -3, -2, 1, 4], [2, 2, -1, -9, 4, -3, 1, 5, -2], [9, -3, 1, 3, -9, -3, 6, 2, 6], [10, -3, 1, -4, 5, 4, -7, -3, -3], [-3, -7, 4, -9, 8, 2, -5, -10, -3], [-1, 1, -9, -13, 2, 2, 4, -3, -1], [6, -7, -4, 0, 11, 4, 2, 0, -7], [-7, -8, 0, 6, -5, 4, 0, 0, 3], [4, 6, 0, -6, -8, 1, -4, -1, -7], [3, 4, 1, 1, -1, 5, -1, 1, -6], [14, -8, 3, -4, -3, 1, 1, 2, 8], [-4, 9, -3, 0, -7, -3, -5, 10, -2], [5, 1, 5, -2, 7, -2, -8, -5, -1], [4, 8, -6, -1, -4, 13, 5, 4, 1], [-2, 5, -4, -7, -11, 5, 1, 1, 0], [5, 6, 6, -3, -7, -6, 3, 1, 6], [4, -8, 9, 6, -5, -1, -5, -2, 2], [-4, -1, -2, -2, 5, -14, -14, -2, -1], [5, 2, 4, -11, -2, 0, -2, -13, 0], [6, 1, -9, -9, 0, 2, 0, 3, 0], [4, -2, 6, -1, 8, -1, -5, 3, -7], [-2, 0, -4, 6, 4, -4, -1, 6, -2], [6, 9, -3, -7, 4, 4, 0, 1, -4], [15, 12, 3, 1, -4, 0, 0, 7, -5], [3, -3, -6, 1, -2, 2, 6, 4, 4], [5, 2, -2, 1, 10, 4, -6, -1, -2], [10, 4, -9, -3, -3, 3, 7, 1, -4], [-1, 5, 14, 4, -17, -6, -1, 1, 3], [29, 2, -3, -4, -8, 3, 0, 7, -1], [9, 8, -3, 2, -16, 9, 0, 7, 9], [21, -4, -5, -6, 16, -1, 2, -5, -3], [-4, 5, -10, -15, 0, -5, -4, -2, -6], [-12, 8, 3, -3, -7, 0, 0, -4, -5], [-10, 2, 9, -5, -3, 6, 5, -9, -3], [-15, -4, 0, -1, -3, -1, 0, -4, 6], [-6, -12, 3, 11, 6, 2, -2, -1, 2], [-5, 5, -1, 12, 5, 0, 0, -5, -3], [2, -6, -2, 8, 6, -4, -6, 0, -6], [-4, 5, -4, -3, -2, 4, 9, -2, -2], [-18, -8, 4, 3, -1, -6, 3, 2, 7], [17, -1, -3, -10, 12, -3, -1, -4, -3], [11, 2, 5, -3, -8, -7, 3, 3, -1], [6, 5, 8, -6, -1, 6, 4, -4, 3], [6, -6, -4, 0, 0, 10, 2, 4, 0], [-17, -9, 11, 14, -5, 4, 5, -3, 8], [7, 2, -1, 9, -6, 2, 3, 0, 0], [1, 1, 7, 4, -1, -4, -6, 6, -1], [-4, 5, -5, 3, -7, 4, 6, 13, 5], [1, -3, 1, -2, -5, 0, 2, 4, -3], [4, -11, 0, 4, 8, -1, 3, -3, 5], [-6, 10, 1, 0, -1, 1, -8, -1, -9], [7, -3, 1, -8, 7, 5, -9, -1, -8], [-8, -9, 9, 2, 4, 1, 3, -1, -5], [13, -1, -1, -10, 4, -9, -1, 4, 4], [2, 8, 5, -5, -9, 1, -5, -2, -3], [-8, 11, -4, 15, -8, 2, 8, 17, 6], [13, 0, -3, -2, 5, 8, 13, -7, 2], [-16, -3, -1, 3, 2, -3, -3, -6, 1], [4, -1, 0, -1, -2, 8, 8, 1, 4], [7, 0, 6, -5, 4, -14, -8, 0, -12], [-16, -5, -1, -5, 1, -5, -4, -9, 8], [-8, 2, -3, -10, -3, -4, -4, -4, 5], [6, 4, 4, -4, -8, -5, 3, -9, 2], [-2, -7, 8, 7, -7, -13, -2, -1, 3], [8, 2, 2, 1, -2, -5, -1, -5, 5], [-7, 1, 2, 4, 16, -1, 1, -3, -1], [21, 1, -3, 2, 6, 3, 7, 4, -11], [6, -9, -10, -10, 8, 0, -4, -4, -2], [-13, -7, 3, -1, 5, -8, -17, 0, -7], [-1, 14, 3, 14, -9, 4, 10, 4, 3], [-26, -7, 1, 1, -7, 9, 0, -4, 6], [10, 8, -12, -2, 0, 3, 8, -3, 1], [14, 7, -4, 8, -1, 14, 4, 2, 1], [1, -6, 4, -8, 5, 4, -2, -3, 2], [-3, 0, -5, -5, 6, -2, -6, 0, 6], [4, -12, 2, 4, 10, -10, -6, -11, 4], [5, -5, 5, -11, 7, 2, -8, -18, -2], [0, 3, 2, -8, 10, -9, -2, -15, -7], [39, 11, 2, 6, -6, 11, 4, -2, 6], [10, 1, -7, -2, 7, 6, 3, -4, 1], [5, 0, 9, 13, 7, -4, -5, 3, -5], [2, 0, -5, 3, 4, 6, 2, 5, -8], [-19, -4, -1, -3, -8, -17, -14, 6, 0], [-3, 6, 5, 4, 3, 3, 4, 7, -2], [16, -2, 9, 2, -3, -18, -11, 2, -7], [5, 4, -6, 2, 4, -1, 3, 5, -3], [6, -1, -4, -7, 7, -1, -2, -2, 7], [10, -1, 6, 15, -3, 10, 2, -8, -2], [-8, 8, 3, 22, 0, 1, 2, 2, -3], [2, -4, 8, 10, 10, 3, 8, -3, 9], [6, 3, 6, 14, 1, 4, 5, -9, 6], [7, -1, -5, 6, 11, -3, 3, 3, -2], [17, 7, 1, -6, -3, 10, -7, -4, -3], [0, -3, -1, -5, 7, -5, -6, -11, 2], [3, -4, -7, 0, 17, 10, 10, -5, 5], [2, 10, 3, -4, -3, 4, -2, -6, -7], [30, 3, 2, 0, -8, -9, 10, 4, 5], [-6, 9, -3, -11, -16, -5, -4, 0, 11], [-3, -2, 2, -8, -5, -5, -3, -1, 8], [-13, -8, -1, 21, 9, 3, -2, -2, 1], [2, 12, -4, 1, -1, 3, 13, -1, 2], [3, 10, -8, 4, -2, -3, 1, 10, 3], [12, -5, -7, 5, 19, -3, 7, 5, -5], [12, -2, 5, 5, 2, 12, 4, 5, -2], [11, -5, -7, -2, 4, 9, 8, -3, -1], [-14, -4, -5, -4, -1, 15, 1, -6, -2], [11, 2, -2, 15, -8, 7, 10, 8, 3], [11, 0, 3, 1, -5, -5, -5, 11, 4], [1, -8, 10, -5, 7, -7, -9, 4, -8], [8, 8, 1, 9, -18, 4, 1, 13, 0], [21, 3, -4, -9, -18, -3, 2, 1, 7], [3, 8, 2, 3, -11, -3, -1, 2, -3], [-5, 6, 4, 15, -6, -3, -3, 8, 7], [-2, -8, 0, 1, 20, -13, -3, -6, -1], [0, 9, 2, 3, 9, -5, 3, -5, -6], [27, 8, 6, 8, -9, 14, 4, -1, 4], [7, 2, -1, -1, 2, -4, -4, -14, -3], [-3, 4, -6, 9, -5, -4, -9, -1, 4], [32, 13, -6, 0, 5, 5, 14, -1, 0], [3, 6, 12, 1, 8, -3, -4, 4, -1], [-15, -12, 4, -3, 8, 2, -3, -13, 0], [-10, 0, 10, 4, -4, 1, -2, 3, 5], [25, 10, -9, -5, -7, 15, 13, 5, -4], [22, 3, 1, -3, -1, 2, 9, -15, -6], [9, 1, 4, 6, -6, 2, 7, 1, -8], [22, -8, -3, -16, 15, 4, 3, -13, -1], [-6, 3, 4, -3, -4, -4, -13, 12, -2], [-10, 2, -9, -3, 1, -3, 6, 3, 8], [-14, -3, -1, 0, -1, 10, -5, -16, 9], [-14, -1, 5, 10, -3, -1, 1, -8, 10], [11, -2, -1, -4, 8, -9, -6, -5, -5], [-26, -1, -2, -8, -15, -4, 2, -1, 11], [4, 2, -9, -6, 7, -1, -3, 2, 2], [18, 0, -3, 4, 7, -15, -9, 5, 2], [4, 1, 0, 8, 1, 7, 0, -7, 4], [-14, -3, -1, -11, 4, 16, -8, -9, 2], [-20, -1, 14, -13, -11, 5, -2, -14, 7], [-5, -9, 15, 14, -11, 2, 3, 1, 8], [33, -10, -8, 6, 2, 5, 3, -5, -1], [18, 9, -5, -10, 10, 2, -2, -4, -4], [-18, -4, 14, 1, -8, 7, 5, -5, -3], [10, 14, -8, -3, 14, 0, -3, 4, -6], [-2, -20, -3, 2, -2, -3, -4, -2, 5], [-4, 0, 8, 17, 2, -3, 8, 11, -2], [14, -11, 3, 0, 4, -14, -9, 9, -2], [-21, -2, 18, 4, -9, -8, -17, 1, -7], [34, 1, -3, -5, 13, 13, 9, -6, -5], [-14, -3, 14, 18, 0, -2, 3, 0, 4], [-19, 0, -6, 1, 8, 10, 6, 3, -4], [-1, -11, -6, -6, 2, 12, 3, -1, 6], [10, 1, 1, -2, 7, -9, 2, -9, 3], [27, 12, 5, 1, -2, -5, 1, 11, -5], [8, 6, -3, -1, -5, -18, -8, 6, 3], [21, -2, 8, -8, -7, -2, 3, 5, 7], [-30, 12, -1, -8, -11, -2, -5, 9, -5], [-13, -8, 7, 5, 1, -2, -2, 0, 3], [18, 5, -12, -8, 8, 17, 0, -3, -1], [-11, -7, 4, -12, -6, -17, -9, 5, -3], [7, 1, -7, -5, -2, -1, 3, 5, -1], [0, 7, 3, -2, -12, -10, -8, 5, 0], [6, 8, -2, 6, -8, 2, -10, -4, -6], [0, -7, 0, -6, 1, 3, 16, -3, 4], [-5, -2, 1, -12, -2, -17, -8, -3, -5], [-4, 7, 12, 1, -14, -2, -4, 2, -10], [-19, 4, 8, -1, -5, 2, -8, -5, -1], [-2, 0, 8, 9, 15, -12, -3, -12, -5], [4, 6, 4, 10, -11, -9, -10, -1, 2], [-10, 10, 3, -1, -5, -1, -2, -13, 2], [0, -2, 5, 1, -6, 10, 2, -1, -2], [17, -11, 1, -3, 13, 0, 1, -2, -7], [-1, -4, 5, 0, -4, -12, -10, 16, -4], [16, -12, -7, -9, 16, -10, -6, -9, 6], [-8, 13, -4, -10, -6, -1, -1, 1, -7], [19, 9, 2, 11, -1, -4, -2, 6, -9], [12, 1, -5, -2, 7, -6, 9, 9, -1], [-8, 13, -3, 1, -9, -9, -2, 4, 8], [6, -13, -4, 2, 11, -1, 10, -5, -6], [2, -10, 8, 0, 9, 5, 3, -2, -2], [-2, -13, -8, -4, 4, 9, -6, 2, -3], [4, -9, -9, -7, 19, -2, -4, -11, 13], [16, 5, -4, -2, 1, -1, 14, 3, 4], [-2, 0, 6, -6, 0, 6, 7, -1, 12], [-2, -5, 0, -3, 4, -5, 10, -11, 5], [-18, -8, 5, 6, 8, 2, 9, -5, 0], [-28, 13, -6, -18, -13, 3, -3, 1, -1], [0, -2, 4, -5, -6, -5, -3, 2, -3], [-11, -16, 4, 12, -7, 0, -5, 2, 9], [16, 13, -8, -7, -1, 12, 9, -9, 7], [14, 8, -10, -18, -12, -5, -8, 10, 1], [-21, -2, 13, -1, -7, -18, -9, -1, -1], [20, 3, -14, 9, 0, 3, -3, 2, 1], [-16, 0, 11, -10, -11, -3, -13, -2, -10], [-13, 5, 10, -7, -9, -10, -2, -1, -3], [5, -15, 5, 8, 19, -7, 8, -7, 6], [-14, 0, -5, -1, 5, -4, -14, 1, 1], [16, -6, 13, -1, -6, -10, 2, 3, 6], [-21, 8, -8, -9, -2, 15, 6, -12, 1], [-13, 9, -10, 2, 6, 18, -1, -6, -2], [11, -6, 4, 20, 12, -4, 5, 4, 2], [0, -2, 11, -7, 11, -2, -9, -5, 2], [17, 0, 2, 1, 4, 8, 7, -1, 6], [-18, -3, -1, -3, 7, -10, 3, 11, -1], [-6, 1, -5, 6, -4, -4, -9, -7, 6], [8, 8, 2, 2, 2, -8, -4, -9, -8], [-5, 2, -12, -12, -5, 12, 19, 3, 5], [17, 1, -3, -11, -7, 4, 11, 0, 5], [18, 11, 1, -13, -12, -7, -7, 0, -8], [13, 16, -5, -6, -8, 22, 24, 6, 0], [0, -9, 10, -7, 1, -10, -3, 2, -1], [-27, 12, 1, -7, -11, 2, -8, 1, -1], [-2, -13, 12, -6, -3, -15, 2, -10, 10], [9, 6, 15, 9, -6, -11, -19, 0, -9], [-15, 8, -9, -13, -10, 7, 11, 1, -1], [-9, -2, 17, -8, -19, -11, -2, -9, 16], [1, 0, -1, 9, 13, -10, -1, 2, 3], [0, -1, -2, -11, 0, -23, -13, 3, -7], [36, 1, -7, 0, -12, -1, -1, 14, 7], [6, -17, 1, 12, 12, 9, 7, -3, 5], [-10, 17, 6, 20, -11, -2, 13, 14, 10], [1, 12, -2, -2, -10, 24, 13, 1, 4], [-16, 2, -1, 3, 6, -5, -9, -4, 9], [-17, -6, 3, 4, -8, 5, -7, 0, 1], [35, -3, -3, -15, 23, -8, -6, -8, -12], [18, 4, -15, -7, 22, 1, 0, 2, -13], [-14, 2, -5, 0, 5, 9, 4, -2, -10], [-8, -5, 12, 17, 1, 5, -7, -5, 2], [-1, 4, -9, -2, -15, -3, -2, 2, -4], [2, 2, 5, -8, -12, -9, -10, -1, -10], [-23, -16, 3, -2, 12, -5, -6, 7, -13], [10, -3, -15, -29, 10, 1, -7, -3, -9], [-14, 4, -1, -14, -7, 7, 12, -3, 5], [13, 10, 8, 16, -10, -1, 11, 8, 3], [8, 14, -6, 13, -19, 18, 11, 17, 13], [-6, 1, -4, -4, -14, 10, 2, 2, 16], [-27, 6, -16, 0, -1, 3, 1, 9, -8], [-16, 3, -11, 2, 8, 4, 0, -8, 10], [22, 7, 4, -2, -13, -3, 12, 2, 8], [-20, -5, 8, -2, -8, -9, -6, 0, -7], [2, -7, -12, -8, 5, 5, 15, -13, 1], [-11, -4, 5, 6, -16, 3, 9, -4, 14], [19, -1, 5, 6, 9, 4, 3, -2, -4], [29, 1, -4, -8, 3, 4, 5, -10, -17], [13, -1, -7, -2, 3, -5, -4, 7, 4], [20, -13, 9, 16, 18, 7, -3, 1, 7], [3, -1, 4, 12, -5, -3, 12, 10, 7], [-21, 2, -12, 8, -1, -1, -1, 8, 4], [11, 6, 2, -9, -14, -8, 4, -1, -6], [-14, -22, 0, -4, 4, -4, 11, -11, 3], [-11, 0, -18, -20, 5, 2, 1, -10, -3], [5, -6, -1, -1, 24, 5, -3, -7, -5], [-1, -11, -9, 2, 11, -3, 8, 11, 6], [-9, -3, 6, 19, -16, 8, 4, 9, 2], [-1, 10, 0, 20, -3, -3, 0, -2, -1], [4, 4, -10, -14, 7, 14, -6, 0, -9], [5, -5, 8, -15, 8, 7, 0, -9, -7], [-27, 0, 10, -5, -15, 11, 9, -3, 8], [14, -2, -12, -10, 6, 0, 8, 11, 0], [-3, 16, -7, 2, 2, 6, 0, 0, -6], [21, 2, -14, -29, 6, -4, -7, -7, -9], [-6, 2, 2, 13, -2, -19, 6, 7, 1], [1, 12, 12, 11, -21, -5, -3, 15, 0], [0, 16, -4, -14, -21, -2, 4, -1, 11], [-7, -10, -9, -4, 18, -10, -16, -8, -12], [12, 3, -9, 0, -6, -1, -3, 18, 3], [-11, 6, 5, -1, -13, 10, 8, 0, 4], [-11, -6, -1, -7, 1, 3, -3, -15, 2], [26, 5, -9, -2, 6, -9, -5, 6, 5], [-2, -2, -8, 16, -21, -6, 5, 11, 3], [18, -8, -1, 2, 4, -5, -4, -10, 4], [40, 9, 1, 4, -3, 8, -7, 6, 5], [4, 26, -2, 0, -14, 16, 22, 8, 4], [-4, -17, 2, 5, 23, -7, -16, -4, -11], [10, 5, 4, -3, 10, 15, 1, 3, -9], [-11, -8, -9, 3, -7, 15, 3, 1, -8], [-15, -7, -5, -15, 18, -14, -26, -2, -10], [24, 2, -13, -6, 15, 11, -1, 2, -6], [-20, -6, 10, 16, -4, -13, 4, 5, -1], [-22, 5, 3, -2, -2, -8, -13, 7, 2], [-3, -10, 0, 10, 4, -1, -1, -3, -13], [-1, 4, -10, -3, -12, -4, -8, 15, -6]];
    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_e();".
// 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_e(:prec:=9)
    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_e();".
// 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_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8004_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-44, 116, 66, -314, 117, 117, -44, -19, 3, 1]>,<7,R![-180, -36, 418, 138, -245, -53, 65, 2, -7, 1]>],Snew);
    return Vf;
end function;