
// Make newform 8004.2.a.g in Magma, downloaded from the LMFDB on 17 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_g();".
// 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_g();".
// 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 := [16, 136, -616, -1156, 3304, 2146, -1714, -697, 347, 80, -31, -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, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [770552, -11305176, 6816868, -2416332, -11289370, 7143630, 3873043, -2165385, -479516, 231201, 19357, -8071], [11033668, 2829120, -63772828, -218775012, -64812155, 111631233, 29877380, -21650127, -3812431, 1862577, 150230, -58412], [-69925000, 65617524, 927980284, -447721968, -890169226, 284027205, 247389829, -67159668, -26540003, 6749571, 958345, -237136], [-32135128, -342835836, 226067380, 2549822760, 913186418, -1259064135, -362660321, 241331934, 44452597, -20607159, -1717169, 643238], [-13046896, -162478008, -18498844, 866406388, 389223760, -426165362, -143477867, 82610957, 17182508, -7166323, -656309, 227217], [-37571512, -445104660, 39670792, 2853762180, 1092840230, -1435974561, -411626960, 280180599, 49854049, -24168090, -1925114, 756551], [21253000, -173254620, -145530924, 1101125008, 547270074, -570735599, -193749385, 114673742, 23105733, -10152007, -886849, 324494], [-33296528, -695410800, 142017764, 3942455436, 1436390248, -2000252604, -547193839, 390702759, 66191846, -33488751, -2548933, 1038679], [-35157824, 341574492, 385348124, -1977984000, -1082368964, 997238067, 366059189, -196470678, -42407701, 17173041, 1592693, -545408], [36924904, -490498764, -366651712, 4256483244, 2066374630, -2179409127, -737802358, 429697791, 87445547, -37462464, -3327328, 1186027]];
    Rf_basisdens := [1, 1, 1181592, 4135572, 16542288, 16542288, 5514096, 16542288, 5514096, 16542288, 8271144, 16542288];
    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, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 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, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [-1, 0, 1, -1, 1, -1, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 1, 0, 1, -1, 1, 1, 0, 0, 1, 1], [-3, 0, 1, 0, 0, -1, 0, 0, -1, 0, 0, 1], [-1, 1, 0, -1, 0, 0, -1, 0, -1, 1, 0, 1], [0, 0, 1, -1, 0, 0, 0, 0, 0, -1, 0, 1], [-1, 0, 1, -1, 1, 0, -1, 0, -1, 0, 0, 0], [1, 1, -1, -1, -1, 1, 1, 1, 0, 0, 0, -1], [-1, 0, 0, -1, 1, 0, 1, 0, 0, 0, -1, -2], [-2, 0, 0, 1, -1, -1, -1, 0, 0, -1, -1, 0], [-1, 1, 2, -2, 1, -1, 1, 0, 1, 1, 1, 1], [1, -1, 1, -2, 0, 0, 0, -2, 0, 0, 0, 1], [-1, 1, -1, 2, -1, 0, 0, 1, 0, 0, 1, 0], [2, 1, -1, 0, 0, 1, -1, 0, 0, 1, 1, -2], [1, -1, 1, -1, -1, -1, -1, 0, 1, 1, 2, 1], [-3, 1, 1, 0, 0, -1, -1, 1, 0, 0, 2, 1], [2, 1, -2, 2, -2, 2, 0, 1, -1, -1, 0, -2], [-1, 1, -1, 1, -1, 0, 1, -1, 0, -1, 0, 0], [5, 1, -1, -2, 0, 2, 0, -1, 0, 1, -3, -1], [-1, 1, 1, -1, 1, -1, 2, -1, 0, 1, 1, 1], [-1, 1, 1, -1, 0, 1, 0, 0, 1, -1, 0, 0], [-3, 0, -2, 1, 0, -1, -1, -1, 0, -1, -2, 0], [1, 1, -1, 0, 1, 2, -2, 1, 0, 0, -1, -2], [0, 1, -3, 0, -1, 4, 1, -1, -1, -1, -3, -1], [-3, 1, -1, -2, 2, 1, 1, -1, -2, 0, -3, -1], [0, -1, 0, 1, 0, 1, -2, -1, -1, 0, -1, 1], [-3, -1, -2, 2, -1, 3, -1, 2, -1, 1, 0, -1], [2, -1, 1, 0, 0, -1, 0, 0, 3, -1, -1, -3], [-3, 4, 2, -2, 3, -2, 2, 0, 1, 2, 2, 1], [2, 0, -3, 2, 1, -1, 0, 0, 0, -2, 0, 0], [-1, 1, 0, 0, 1, -1, 2, 0, 0, -2, 0, 1], [3, -1, -2, -2, 0, 0, 2, 0, 2, 0, 1, 1], [-5, -2, 2, 1, 1, -5, -1, 1, 1, -2, 0, 1], [-2, 1, 0, 1, 1, 2, 0, -1, -1, 0, -2, -1], [-2, -2, 2, -2, -1, 3, -2, 0, 0, 1, 1, 1], [-2, -2, 1, 2, -1, -2, 0, 1, 1, 0, 1, 1], [-3, -1, -4, 3, -3, 4, -2, 0, -2, 1, -2, -2], [2, 3, -2, 3, 2, 1, 1, 1, -1, -1, 0, 1], [-7, -2, 4, 0, 2, -4, 0, 1, 2, -1, 2, 1], [0, -2, 3, 2, -2, -4, -3, 1, 2, 2, 2, 0], [0, -1, -2, 1, -1, 3, 0, 2, -2, 2, 0, -4], [-5, -1, 2, -3, 3, -3, -1, -2, 3, 0, 1, 2], [-7, 0, -3, 3, -1, 2, -2, 1, -1, 0, 1, 2], [1, -3, 0, 3, -2, 2, -1, 0, -2, 1, -1, -1], [-4, -1, 2, 2, 2, 1, 1, 1, 1, -1, 3, -1], [-4, 0, -3, 3, -1, 3, 1, 0, 0, 0, -1, -1], [-3, 0, 3, -2, 0, 1, 1, -1, 1, 1, 0, 0], [-2, -2, -3, -1, -1, 1, -1, -2, -4, -1, -6, 0], [-2, 2, -5, 1, -2, 4, 3, 0, -2, -1, -4, -4], [4, 2, 1, -2, -1, 3, 1, 3, -1, 0, 1, 0], [-2, -2, 2, 0, 1, -3, 0, -2, -1, -2, 1, 0], [-1, 2, 0, 3, 0, 2, 1, 2, 1, -1, -2, -3], [4, 2, -2, -1, -2, 0, 2, 4, 2, 0, 1, 0], [1, 0, -2, -2, -3, 5, -1, 1, -2, 1, -3, 0], [-3, -2, 4, 0, 4, -3, 2, -2, 3, -1, 1, 2], [2, -6, -2, 2, -2, 2, -1, -1, 1, 0, 0, -4], [1, 2, -1, 2, -1, 4, 2, 4, -3, 1, 2, -2], [-3, 3, 1, -1, 1, -2, 2, -2, 1, -1, 3, -2], [2, 1, 0, -1, -3, 1, 1, 1, -1, 0, -3, 0], [-1, -3, 0, 1, 0, 0, -3, 0, -1, 0, -4, -2], [3, -6, 0, -1, -3, -1, 1, 1, 1, 2, 2, 3], [-4, -1, -1, 0, -1, 0, -1, -1, -2, -1, -3, -3], [-4, 2, -2, 2, -1, 4, 0, 1, 1, 0, -2, 1], [-4, -3, 6, -3, 2, -7, -4, -3, 1, 1, 0, 4], [3, 1, -6, -1, 0, 3, 1, -2, -1, 0, -1, 0], [-1, -1, 2, -4, 2, 2, 0, 2, 1, 0, 2, -1], [0, 0, -2, 0, -3, 7, -1, 0, -4, 2, 0, -1], [0, -1, 1, -3, -2, 1, -2, -1, 1, 2, 1, 1], [-13, -3, -3, 3, -2, 2, -5, -1, 0, -1, 0, -2], [-2, 1, -4, 2, -1, 4, -2, -2, -4, -1, -2, -3], [-3, 2, 4, -1, 0, -3, 1, 1, 4, -1, 5, 0], [-5, 0, 3, -1, 2, -2, 1, 1, 0, 2, 3, 3], [-8, -3, 3, 4, 3, -2, -1, -1, 1, -3, 2, 1], [-7, -5, -1, 1, -2, -2, -2, -1, 0, 2, 0, 2], [2, 5, -3, 1, -2, 4, 0, 2, 2, 0, -2, -4], [-2, 2, -2, 5, 4, 1, 0, 2, 0, -1, -2, -3], [-2, -6, 1, 0, 0, -4, -2, -2, -1, 2, 3, 0], [-11, -2, 0, 0, 1, 0, 0, -2, 1, -1, 1, -2], [-6, -1, 5, -3, 2, -4, -1, -3, 5, 2, 0, 0], [1, 4, 2, -5, 3, -1, 2, -2, -1, 2, -3, -1], [-4, 2, 0, -4, 0, -1, 4, -1, -1, 0, -1, -1], [0, -1, -5, -1, -3, -1, 0, 2, -1, -2, -1, -4], [-6, 0, -1, 1, -1, -1, 1, 4, 2, -2, -2, 1], [-4, 2, -2, 5, -2, 0, -1, 0, 0, -1, -2, 0], [-4, -1, -3, 3, -5, 8, -1, 0, -6, 0, -3, -3], [-7, -2, 4, 0, -3, 1, -1, -4, -3, 3, -4, 1], [-6, 0, -6, 2, 1, 2, 1, -4, -1, -2, -3, -1], [-4, -4, 1, -1, 1, 1, 0, -1, -5, 0, -2, -3], [-5, 6, -4, -2, 1, 7, 1, -2, -4, 1, -1, -1], [-2, -6, -3, 0, -3, -2, 1, -3, 3, -3, 0, 3], [2, -1, 1, 1, -3, 2, -3, 3, 1, 3, 2, -1], [-8, -1, 0, -4, 2, -1, -1, -1, 3, -1, -4, 3], [4, 0, -5, 5, -3, -1, 0, 3, -2, -1, -3, 2], [-8, 1, 1, 0, 3, -2, 2, 1, -1, 2, 3, 3], [-2, 4, -2, -2, -1, 2, 3, 4, 2, -1, 2, -4], [-1, 6, 0, 0, 4, -1, 3, 4, 3, 0, 3, -4], [0, -4, -2, 7, -2, 3, -3, 0, 0, 0, 1, -3], [-7, -6, 4, -1, -3, 0, -5, -2, -1, 2, -2, 2], [-1, -5, -2, 10, -3, -1, -4, 2, 0, -2, 2, 1], [3, 4, -5, -1, 0, 3, 1, -2, 2, 0, -5, -4], [-1, 6, 2, -5, 2, -2, 5, 1, 0, 0, 1, -1], [-6, -4, 2, -4, 0, 0, -1, 3, -3, 0, 1, 0], [7, 3, -1, 1, 0, 2, -1, 6, 1, -1, 7, -1], [0, 1, -2, -2, 0, 5, 6, 0, -2, 1, 2, 1], [-5, -3, 3, 4, -3, -2, -5, 1, 2, 3, 4, 2], [2, 5, -5, -1, -2, 9, 1, 1, -3, -2, -4, -2], [1, 4, -4, 4, -1, 3, 0, 2, 1, 0, 2, 3], [-15, -3, 7, -2, 5, -5, -2, -3, 2, 0, 1, -2], [13, -2, -3, -3, -5, 4, -1, -1, 2, -1, -2, 0], [-3, 0, -4, 0, -3, 3, -1, 0, -2, 3, 3, -1], [-3, 0, 5, 3, 1, -6, 1, 2, 5, 0, 8, 6], [-2, -5, 6, -6, 2, -6, 1, 2, 4, 0, 0, 3], [-3, 1, -8, 5, -1, 5, 2, 1, -2, 1, -1, -3], [4, 1, -3, -4, -5, 4, -1, 1, -2, 1, -2, 3], [1, 4, 2, -1, -1, 0, 1, -1, -2, 3, -1, 6], [-4, 1, -3, 1, -2, 1, -2, 3, -1, 2, 1, -3], [8, 3, -2, -3, 2, 2, 0, -2, -2, -2, -4, 2], [-12, -2, -3, 3, -4, 3, 2, 3, -3, 2, 1, 1], [-3, 8, -2, 1, 1, -3, 5, 3, 4, -2, 1, 1], [-5, 2, -1, -1, 2, -6, 4, 0, 2, -2, 1, 1], [9, -7, 3, 1, 2, -7, 0, -1, 0, -4, -1, -2], [-6, -2, 8, -6, 4, -7, 2, -3, 1, -2, 5, 5], [-2, 1, -4, -6, 0, 4, 1, -4, -6, 1, -7, -4], [10, 4, -7, -1, -2, 3, 0, 2, 2, -2, 2, -2], [-12, 9, 7, -9, 5, -1, 2, -3, -1, 0, -5, 1], [-11, 0, 5, 0, 4, -7, 1, -1, 6, 2, 5, 3], [3, -3, -5, 1, -1, 2, 0, 0, -6, -1, -3, 0], [13, 4, -6, 2, 1, 5, 3, -2, 1, 2, -6, -2], [-5, 1, -6, -2, -2, 9, -4, 0, -5, -1, -5, -6], [5, -4, -3, 1, -5, 2, -7, 1, -2, 1, -5, -4], [-5, -3, 3, -1, -2, 1, -1, 2, 1, -2, 0, -2], [-12, -2, 3, 0, 1, -1, -1, 5, 1, 0, 6, 0], [-2, -3, 5, -3, 1, -5, -2, -4, 2, -2, 0, 4], [-5, 1, 7, -4, 3, -2, 0, 1, -1, 4, 2, -2], [15, 3, -6, 1, -3, 6, 1, -1, -4, -1, -8, -4], [2, -2, 2, 5, -1, 2, 1, 3, 1, 0, 2, -4], [-8, 3, 0, -2, 4, -3, 1, 3, 1, 3, 2, 1], [0, 5, -7, 1, 0, 3, -3, 0, -8, 0, -3, -1], [-2, 2, -4, 3, -1, 0, 2, 2, -4, -1, -1, -3], [-4, 4, -1, 1, 0, 1, -3, -1, -3, 1, -4, 1], [11, -5, 5, -1, 0, -5, -2, -4, 3, 3, 3, -2], [4, 3, 4, -5, 1, -4, 3, -4, 2, -2, -4, -3], [-2, -2, 7, -8, -1, 1, 3, 3, 0, 2, 4, -2], [-5, 0, -2, 0, -4, 2, -4, -2, -3, 3, -3, 2], [2, -7, 4, -1, -1, 5, 0, -2, -1, 2, -1, -4], [3, -1, -10, 10, -5, 2, -2, -1, -6, -2, -5, 0], [10, 3, 4, -8, 5, -3, 4, -4, 0, 1, 5, 1], [8, 4, 2, -10, 5, 4, 1, -6, 2, -1, -8, -2], [15, 0, -1, 6, 2, -4, 4, 0, 3, -4, 0, -1], [9, 3, 0, 7, -2, 1, 4, 0, 2, 1, -1, 0], [2, 1, 3, -2, -3, 3, 1, 2, -6, 1, 4, 6], [-9, 4, 2, -2, 1, -5, 4, 2, -1, 0, 1, 1], [-14, 6, 7, -5, 6, -4, 3, 0, 0, 5, 4, 6], [2, -1, -6, 5, 1, -1, -2, -3, 3, -3, -2, -2], [15, -3, 4, -8, 2, 0, -1, -3, 6, -2, 1, -3], [-10, 4, 0, 10, -3, 1, 1, 5, 5, 0, 2, -2], [2, -2, 7, 0, 1, -7, 1, -4, 6, -3, 3, 0], [-6, 7, 7, -2, 5, -5, 0, -5, 1, 2, 2, 3], [0, -2, 3, 1, 6, -4, 1, 0, 0, 5, 0, 0], [2, 1, 1, 5, -4, -3, -2, 7, -1, -2, 7, -1], [-10, -7, 2, 6, 0, 1, 0, 0, -2, 5, 0, 1], [3, -7, -3, 2, -2, -1, -1, -4, 1, -4, -6, -4], [-3, 4, -3, -4, 2, 4, 0, -7, -6, -3, -3, -1], [-6, -8, 8, -3, 0, -10, -4, -3, 2, -3, -1, 6], [-9, 3, 4, 4, 3, -6, 0, 1, 2, 4, 4, 10], [3, -4, 6, -3, 1, -3, 1, 3, 2, -1, -1, 0], [-2, 3, 5, 1, -3, -7, -7, 3, -1, 2, 4, 4], [3, -4, 6, 3, -2, 2, 0, 5, 2, 3, 5, -1], [2, -1, 0, 2, -4, 8, 3, 2, -3, 3, 1, 2], [-17, 3, 0, 1, 4, -8, 0, 1, 4, -1, 0, -1], [-1, 4, 1, -6, 8, -2, 3, -5, 0, -4, 0, 1], [2, -3, 7, -5, 8, -1, 5, 4, 4, -2, -1, 3], [-2, 3, 8, -5, 3, -8, -4, 1, 5, 0, 3, 5], [10, 3, -2, -3, -2, -4, -1, -3, 1, 1, 1, 1], [2, -8, 5, 3, -1, -9, -4, 3, 5, 3, 6, 8], [15, 2, -2, -2, -4, 4, -1, 0, 0, -5, -2, 1], [3, -7, -3, 8, -8, 3, -4, 1, 0, -2, -2, 0], [-1, -6, 6, 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-7, -5, -3, -2, 9], [26, 8, -6, -9, 1, 12, 5, -1, -11, 5, -8, -5], [-2, -3, -2, -4, 2, -5, -3, -7, -7, 1, -5, -1], [0, 6, 4, 5, 10, -4, 9, -1, 1, -2, 3, 2], [-13, 0, -9, -3, -7, 1, 0, -8, 1, -5, -7, -4], [-3, 2, -4, 0, 1, -6, -3, -1, 8, 1, -3, 2], [10, -14, 1, -6, 4, 0, 1, 1, 2, 6, 0, -2], [-3, 3, -4, -8, -3, 6, 0, -7, -9, -4, -1, -2], [-11, -8, -8, 13, -8, 4, -4, 3, -6, 0, -6, 2], [-27, -1, 9, -4, 2, -7, -7, -7, 0, 1, -2, 6], [-26, 12, 7, 0, -2, -2, 1, 7, 0, 6, 13, 0], [-1, 1, -5, 5, -9, -2, -3, -1, 4, 9, 6, 0], [-2, -6, -2, 3, 0, -1, 5, -1, 1, -1, 8, -5], [1, 0, -11, 7, -8, 19, 0, 9, -2, -1, -7, -11], [21, -8, -1, 14, -6, 0, -1, 7, -4, -2, 4, -3], [16, -10, 2, -7, -9, 2, -6, -2, 5, -1, 2, 3], [9, -4, -4, 2, -2, 10, -2, 2, -5, 1, -14, -6], [0, -4, 5, -9, 10, -6, -2, -10, 6, 4, 4, -4], [-9, -4, -8, 0, 4, 2, 8, -3, -2, -3, -2, -7], [19, -6, -6, 7, -8, 7, -12, -1, -8, -6, -11, 2], [2, -9, -2, 13, 0, -12, 2, 5, 3, -3, 9, 0], [5, -6, 7, -9, 7, -12, -10, -11, 7, -2, -11, 2], [-11, -6, -2, -7, 1, 6, -1, 0, -3, 2, -4, -8], [12, 11, -15, 7, -5, 20, 5, 2, -6, 4, -8, -11], [0, 0, 2, 6, -2, -15, -7, -7, 1, 4, 6, 18], [-24, 0, 4, 1, 6, 10, 4, -3, -1, -2, -2, 1], [-21, 0, -9, 0, 6, 1, -6, -4, -5, -10, -12, -5], [12, 7, 2, -7, 0, 0, 6, 0, 4, 8, 7, 4], [-38, -6, 0, 10, 2, -6, -5, -5, -11, -2, 2, 6], [-7, 1, 14, -26, 14, -11, 9, -10, 13, 4, 4, 2], [10, 1, -10, 5, -11, 20, 4, -1, -7, 1, -6, -14], [4, -16, -1, 8, -5, -12, -7, 4, 4, 0, 11, 7], [-23, 2, -10, 11, -2, -2, 6, 9, 6, 1, 1, -5], [7, -4, -8, 3, -1, -3, -4, -3, 0, 4, 4, 12], [-30, 4, 13, 1, 7, -13, 7, 5, 1, 2, 1, 4], [-21, -6, 10, -3, 6, 3, 1, 2, -7, 2, 2, 2], [-10, 4, 3, 4, 7, 1, 0, -6, -2, 2, 2, -4], [-5, 7, -2, 13, 4, 2, 4, 3, -4, 5, -1, -3], [-24, -6, 15, -13, 14, -15, -3, -16, 0, 3, -2, 8], [18, 4, -1, 1, 0, -3, 6, -5, 1, -4, 9, 5], [-2, 2, 1, 7, -5, 3, -5, -2, -8, 3, 4, 4], [26, -21, 15, -13, 2, -5, -2, -5, 8, 0, 6, 3], [-5, 10, -2, 0, -4, 4, 0, -3, 2, 1, 1, 1], [-2, 12, 5, -5, 12, -5, -5, -5, -5, 3, 5, 5], [10, 5, 4, -13, 5, 8, 8, 0, 0, -2, -7, -4], [1, -4, 4, 1, 1, 5, 7, 3, 1, 6, 6, -3], [18, -7, -10, -3, -9, 10, -7, -2, -5, 4, -4, -1], [6, 9, 3, 10, 10, -2, 0, 3, 3, -1, -1, 2], [-20, -2, 8, 5, -3, -14, 0, 2, 0, -3, 7, 5], [9, 4, 6, -7, -2, -8, -1, 6, 5, 3, 14, 11], [5, -11, -5, 1, 7, -1, -3, -6, 7, -6, -8, -12], [22, -3, -8, 13, 1, 4, 0, -1, 3, -2, 2, -11], [-10, 9, 4, -12, 6, 5, 3, -7, -13, 5, -3, 5], [20, 3, -11, 9, -8, 2, -2, 2, 7, -2, -2, -10], [1, 3, -6, 4, -7, 3, 9, 0, -3, -2, -1, 2], [-10, -1, -4, 4, -5, 4, -2, 8, 0, 5, -4, 5], [9, 3, -13, 9, -13, 4, 1, 4, -5, 2, -7, -6], [-37, 4, 5, -5, 0, -6, -3, 3, 0, -3, 11, 6], [-4, -4, -12, 13, -9, 4, -3, 15, -1, -3, -2, -5], [20, 5, -9, -9, -2, 5, 9, 2, -8, 2, -8, 1], [24, -5, -6, 1, -8, 8, -13, -9, -8, 2, -8, -2], [8, -18, 10, -9, 5, -13, -2, -7, 5, 1, 6, 14], [-12, -13, 2, 6, 2, -6, 0, -7, 5, 3, -4, 0], [30, -8, -10, -6, -6, 13, -1, -4, 0, -6, -13, -11], [0, 9, -2, -4, -1, 16, 7, 1, -7, -1, 4, 3], [0, 3, 3, -2, 11, -4, 5, 7, 7, 3, 16, 3]];
    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_g();".
// 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_g(:prec:=12)
    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_g();".
// 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_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8004_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![16, -136, -616, 1156, 3304, -2146, -1714, 697, 347, -80, -31, 3, 1]>,<7,R![-56, -1704, -10516, -7720, 7026, 5494, -2177, -1320, 382, 127, -33, -4, 1]>],Snew);
    return Vf;
end function;