
// Make newform 8004.2.a.f in Magma, downloaded from the LMFDB on 24 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_f();".
// 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_f();".
// 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 := [24, 60, -26, -118, 1, 75, 4, -17, -1, 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, 1, 0, 0, 0, 0, 0, 0, 0], [-182, 258, 553, -480, -559, 161, 180, -2, -11], [-41, 278, 307, -382, -302, 118, 88, -4, -5], [-162, -104, 895, 252, -1061, -113, 322, 24, -21], [24, -74, -749, 229, 1050, -67, -343, -13, 22], [-297, -559, 193, 683, 109, -210, -67, 10, 4], [-702, -598, 1861, 854, -1447, -297, 364, 36, -23], [-495, -484, 1319, 685, -1014, -248, 251, 28, -16]];
    Rf_basisdens := [1, 1, 34, 17, 34, 34, 17, 34, 17];
    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, 0, 0, 0, -1, 0, 0], [-1, 0, 0, -1, 0, 0, 0, 0, 0], [0, 1, 0, 1, 0, 0, 1, -1, 0], [0, 1, 0, 0, 0, 0, 0, 1, 0], [-1, -1, 0, 0, -1, -1, 0, 1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, -1, 1, 1, 0, 1, 0], [1, 0, -2, 0, -1, 0, 1, 0, 0], [-3, 1, 2, -1, 0, 1, -2, -1, 1], [-1, -1, 0, 0, 0, 1, 1, 0, -1], [-2, 2, 0, 1, -1, 1, 0, -2, 2], [1, 1, 0, 1, 1, -1, 2, 2, -1], [-1, 2, -1, 1, -1, 1, 0, -2, 1], [0, 0, -1, 0, 2, 0, 2, 1, -2], [-3, 1, 1, -1, 1, -1, 0, 1, 1], [3, 2, 0, 3, 1, -1, 2, -1, -3], [-5, -1, 3, -4, -2, -1, -4, 0, 0], [1, 0, -1, 1, 3, 1, 1, 0, 2], [3, 0, -2, 4, 1, -1, 4, 0, -2], [-3, 2, -2, 1, 0, -2, 1, -1, 1], [-4, -2, -2, 1, 0, 0, 0, 0, 1], [-4, 0, 2, -2, -1, -1, -2, 0, 1], [-3, 1, 1, -1, -1, -1, -2, -1, -1], [1, 2, -1, 3, 0, -3, 1, 0, -2], [-3, -2, 2, -3, -1, -1, 0, 4, -1], [-2, -2, 0, 1, 2, 1, -2, 1, 0], [-1, -3, 1, -2, -2, 1, 0, 2, 2], [-3, -1, 1, 1, 0, 2, 1, -1, 1], [-9, 1, 3, -3, -2, 0, -6, -2, 2], [0, 0, -3, 4, -2, -2, 0, -1, -2], [-1, -2, 0, -1, -1, -3, -1, 4, -2], [-4, 2, 1, 1, -2, 0, 0, -2, -1], [3, -1, -2, -1, 2, 0, 1, -1, 1], [-2, -1, 0, -1, 1, 4, 0, -3, 1], [1, -3, 0, -2, 4, 1, 4, 2, -2], [0, -3, -3, 0, -1, 1, -1, 0, 0], [2, 3, -2, 3, -1, -3, 1, -1, -1], [-5, -2, -2, 0, -5, -2, -1, -3, 2], [0, -1, -1, -1, -1, 3, -1, -2, 1], [-3, 0, -1, -3, 3, 2, 3, -2, -1], [-1, 4, 0, -1, 2, 3, -1, -4, 4], [-4, 1, 0, 0, 1, 0, 1, 1, -1], [0, -5, -2, 2, 1, -1, 3, 2, 0], [1, 0, 6, -1, 3, 0, -1, 2, 1], [2, -2, -2, 0, -1, 2, 1, -3, 1], [-3, -2, -1, 0, 3, -2, 1, 3, -6], [-2, -1, 1, -1, 2, 1, -4, 1, -2], [9, -3, -1, -4, 4, 3, 0, 1, -2], [-5, 2, 1, 0, 3, 6, -1, -2, 2], [2, -7, -1, -2, 0, -1, 2, -1, -3], [0, -3, 4, 1, 0, 1, 0, 2, -2], [1, 1, 2, -3, 0, 0, -3, -2, 1], [0, -2, 0, -3, -3, 2, -3, 5, -2], [0, -1, 1, 2, 0, -2, 1, 1, 1], [-5, -3, 3, -1, 3, 4, 0, -1, 2], [-5, 2, -4, 3, -4, -1, -4, -5, 1], [3, 1, -1, 0, 2, -2, -4, -1, -1], [2, -2, -3, -1, 3, 0, 0, 5, -3], [7, 0, -5, 0, 1, -1, 4, -1, -2], [10, 1, 0, 1, 3, -4, 4, 5, -3], [-17, -4, 8, -5, 2, 3, -3, 1, 0], [-4, 0, -1, 1, -2, 1, -1, 2, -1], [2, -4, 1, 1, 4, 2, 4, 3, -3], [-18, 6, 0, -2, -2, 2, -8, -3, 4], [-14, -3, 4, -1, -5, 1, -5, -1, 5], [-5, 6, -5, 6, -4, -2, 0, -4, 5], [2, -3, 0, 2, 3, 0, 2, 3, 0], [-1, 0, -2, 3, -4, 1, -1, 0, 2], [-6, -3, 1, 2, -2, -1, 1, 2, 0], [-10, 5, 4, 1, -5, -3, -3, -6, 3], [2, -4, -4, 5, -2, 1, 2, 2, 0], [9, 4, -9, 3, 4, -4, 8, 1, -4], [-5, -5, 4, -4, 0, 2, -4, -1, 1], [2, -7, 2, -1, -1, 0, -2, 1, -3], [10, -5, -3, 1, 0, 1, 1, 0, -7], [2, 2, -1, 5, 4, -1, 7, 2, -3], [-5, -2, 0, 1, -4, -2, 0, 0, 2], [-1, -1, 2, 2, -3, -2, 0, 0, -3], [-4, 7, 6, -1, 2, 2, -3, -4, 6], [2, -3, -4, -1, -2, -6, 3, 6, -6], [-1, 6, -3, 0, 4, 2, 2, -6, 1], [-2, -5, 3, -2, -4, -1, -2, 0, 5], [-2, 0, 2, 0, -3, -2, -3, 3, 3], [8, 0, 1, 1, 6, -1, 5, 0, -5], [5, -3, -3, 8, 0, -2, 3, 1, -4], [-5, 2, -3, 2, -3, -4, -1, -3, 2], [-4, -6, 0, -4, -4, 1, -7, -2, 2], [1, 4, 0, -2, 3, -2, 1, 1, 4], [6, 0, -9, 4, 4, -4, 8, 1, -6], [-3, 7, 0, -3, 1, 3, 3, -4, 2], [-12, 2, 7, -4, -6, 2, -12, -2, 4], [-2, -5, 6, -5, -3, -1, -3, 1, -3], [9, 7, -7, 8, -1, -1, 3, -2, -1], [-1, -1, -2, -3, 2, 2, 2, 0, 0], [-18, -6, 0, -5, -4, 2, -5, -4, 4], [3, -3, -2, 2, 5, 4, 8, 1, 1], [-17, 6, 3, -1, -1, 0, 1, -5, 9], [7, -1, -1, 3, -3, -8, 5, 6, -5], [1, 1, 4, -4, 0, 5, 2, 2, 4], [-7, 6, 7, -6, 4, -1, -6, 0, 2], [-2, 4, 1, -3, 1, 2, -6, -2, 5], [-13, 8, 0, 6, -7, -1, -8, -2, 4], [21, -3, -2, 4, -2, -7, 6, 4, -6], [3, -1, -1, 2, 6, 5, 12, 1, 2], [-1, -4, -1, -2, 2, -1, -1, -4, 3], [10, -6, 1, 3, -1, 4, 3, 2, -4], [-10, 4, 3, 0, -2, -7, -5, 6, -2], [-7, 6, 4, -5, 2, 6, -4, -2, 4], [9, -8, 4, -2, 5, 1, 3, 6, -3], [6, 4, -2, 2, 5, -1, 1, 3, -6], [-1, 5, 0, 3, 0, 3, -1, -5, 6], [-3, -5, -5, 10, 0, 0, 8, -2, -1], [-4, 3, -2, -2, -1, 1, -4, -2, 1], [-5, 5, 9, -8, -3, 5, -9, -2, 3], [2, 2, 4, -9, -1, -3, -4, -2, 0], [-11, -6, 4, -7, -3, 1, -6, -1, -3], [-8, 10, -3, 2, -2, -5, 3, -2, -2], [-13, -3, 6, -6, -8, 4, -11, -3, 4], [-7, -2, 1, 3, -5, 1, -3, -9, 8], [19, 2, 1, 5, 6, -4, 10, 7, -6], [-11, 3, 8, -2, 4, -2, -1, -4, 0], [-7, 13, -1, 7, -6, -4, -6, -5, 6], [-10, 5, 3, 0, -3, 5, -5, -7, 12], [0, -8, -5, 1, 1, 5, -4, -8, 2], [-5, -4, 3, -10, -3, -5, -5, 5, -1], [3, 5, 3, -2, -4, -4, -3, -3, 2], [2, -4, 1, 4, 7, 4, 7, 8, -7], [0, 2, 4, -11, -4, 1, -11, 0, 3], [-10, 11, 3, -3, 3, 3, -8, -4, 4], [-5, -8, 1, 3, 0, -2, 8, 2, 2], [4, -3, 3, 4, 4, -1, 4, -1, -1], [-4, 4, 2, -4, 1, -6, -5, 4, 3], [1, 3, -3, 4, -5, 1, 8, -5, 4], [-6, 8, -3, -2, 3, 7, 3, -1, 0], [-11, -6, 2, -3, -2, 1, -4, 5, 5], [0, 2, -2, -2, -4, 0, -8, -2, 0], [1, -6, -4, 1, 5, 0, 1, 3, -1], [15, 1, -8, 1, 6, -3, 12, 4, -3], [7, -6, -2, 3, -1, -1, 6, 8, -1], [6, -2, -8, 7, 0, -1, 7, 6, -1], [1, 5, -4, -2, -3, -2, -2, 5, -5], [1, 1, -3, 4, -1, -8, 2, 6, -7], [-1, 5, -7, 5, -7, -4, 6, -6, -4], [7, -8, -10, 9, -1, -6, 7, 1, -5], [-9, -1, 1, -7, -1, 1, -5, 13, 0], [17, -5, 4, -3, -4, -2, -5, 6, 1], [14, -9, 0, -3, 4, -2, -5, 7, -4], [9, 8, -3, 0, 3, 0, 0, 5, -3], [6, 8, 1, -1, -1, 0, 5, -1, 0], [12, 5, -1, -7, 4, 1, 4, 2, 4], [8, 8, -3, 9, -5, -9, 3, 4, -3], [23, -1, 2, 4, 2, -5, 6, 8, -4], [1, 8, 4, 6, -7, -1, -3, 4, 1], [-12, 2, -3, -3, -3, -3, 0, -6, -2], [-12, -4, -6, 1, -4, -2, -2, -4, -1], [9, -4, -1, -1, -2, 1, 0, -8, -3], [12, -10, 0, -3, 1, 8, 5, 3, 8], [6, -11, -8, 7, -5, -6, 3, -1, 2], [0, -7, 5, -12, -2, 3, -16, 4, -1], [11, 2, 2, 1, 10, 8, 6, -4, 4], [-16, -2, 11, -11, 0, 6, -9, -1, 2], [-5, 6, 2, -9, 3, 7, -4, 5, 1], [2, 2, 0, 9, -1, -2, -3, 5, -2], [1, -8, 1, -9, -4, 4, -6, 5, -2], [-1, 4, 6, -3, 10, 10, 6, 0, 2], [10, 6, -8, 5, 6, -3, 5, -5, -5], [-1, -4, 1, -3, -3, -3, -10, 7, 3], [5, 5, 5, -4, 2, 4, -1, 0, 12], [2, 0, 8, -5, -2, 3, -1, 5, 3], [5, -8, -5, 1, 1, 2, 11, 1, -3], [-1, 5, 4, -2, 0, 2, -3, 0, -4], [-17, 0, 9, -2, 1, 3, -13, 3, -1], [-6, 3, -5, 5, -7, -2, -12, -5, -1], [-5, 2, 10, -4, 3, 3, -4, 6, 0], [5, -1, -2, 1, 2, -5, 7, 3, 0], [-11, -3, -11, 12, -5, -6, 4, -4, -3], [-21, -1, 5, -6, -5, 0, -14, -4, 3], [12, 2, -1, 2, 1, -3, -4, -6, 3], [15, 2, -6, 10, 3, -6, 7, 1, -12], [-4, -1, 4, 0, -6, -3, -2, 2, 3], [-12, 3, 8, -10, -3, -1, -15, 2, 2], [-8, -3, 4, 5, -2, 5, -7, -7, 7], [18, -1, -9, 4, 6, 5, 6, -4, -1], [8, -4, -7, 5, 9, -1, 13, 1, -7], [-3, -1, 0, 7, 2, -4, 7, 10, -3], [-2, -7, 5, 0, -6, 8, -9, -5, 3], [2, 13, 2, -1, -9, -2, -8, -6, 7], [14, -9, 2, 4, -4, -5, 4, 11, -7], [-3, 4, 6, -2, -9, -5, 1, 0, 3], [-4, 0, 8, -8, 0, 12, -4, -2, 10], [11, -4, -7, 4, 6, 9, 9, -1, -3], [-12, -13, -1, -10, 3, 2, -5, -2, 3], [-4, 11, -2, 4, -9, 0, 4, -13, 8], [-11, 1, -4, 7, -1, 8, -2, -9, 8], [10, -3, -3, 1, 12, 1, 11, -2, 3], [21, 8, -2, -2, 1, 5, -2, -5, -4], [-11, -5, -4, 0, 9, -2, 9, 6, -4], [13, 2, 3, -1, -9, 2, 1, -3, 1], [-21, -10, 2, 0, -2, 5, -7, -11, 3], [4, -1, 5, 1, 3, -4, 8, 11, -1], [8, 9, -11, 9, -4, -9, 0, -5, -4], [-10, 14, 8, -5, 3, 1, -10, -3, 4], [-15, 3, 9, -2, -6, 3, -3, -9, 5], [7, -3, 7, -4, 0, 3, -4, 5, 2], [0, -10, 6, 3, 3, 1, 2, -3, -6], [-1, 6, 1, -5, -1, -5, -12, -5, 1], [15, -1, -11, 8, -6, -9, 16, -4, -4], [37, -1, -8, 14, 4, -3, 20, 0, -8], [-3, -1, -5, 8, 1, -3, 6, 0, -2], [7, -5, -6, 4, -1, 1, 5, 5, 1], [-5, 10, 5, -3, -3, -5, -4, -7, -5], [-10, 5, 5, -7, -7, -1, -7, -3, 1], [0, -9, 2, -9, -1, 2, -12, 4, 3], [4, 1, 1, -1, 9, -1, 5, 7, -3], [-7, -7, -3, 2, 0, 3, 4, -6, 6], [14, -2, 1, -3, -2, -1, 5, 2, -7], [-3, 3, -2, -4, 5, -4, -2, 9, 3], [12, -6, 3, -8, 7, 5, -11, -3, 4], [-17, -2, 1, 5, 8, 4, 8, 7, -6], [22, 4, -4, 4, -3, -9, 13, 5, -4], [13, -12, 13, -13, 4, 7, -7, 9, 2], [-2, -2, -7, -2, -9, 1, 0, -7, 5], [0, 8, 0, -9, -1, -6, 2, 2, -1], [6, -11, 3, -6, -11, -5, -17, -1, 8], [-7, -4, -3, 0, 1, 3, 5, 0, 3], [10, -5, -6, 1, 3, 4, 4, 5, -1], [-1, -8, 3, -9, -6, -3, -15, 6, -4], [-6, 1, 10, 5, 4, -1, 1, 1, -3], [3, -16, -6, 1, 0, 7, 10, -2, -1], [2, -8, 3, -4, 0, 7, 7, -2, 6], [-30, -4, -2, -1, 1, 6, 0, -5, 1], [20, -4, 2, 4, -2, -2, 4, 4, -8], [-7, -5, -1, 2, 2, 4, -5, 0, 0], [1, -1, -1, 0, -4, -10, 0, 4, -1], [-12, -10, 7, -11, -10, -7, -13, 5, 3], [0, 8, 0, 9, 0, 4, -6, -18, 13], [4, -5, 4, -7, -9, -9, -3, 9, -7], [-1, -6, -1, -7, -11, 6, -3, -3, 1], [15, 10, -11, 14, 1, 4, 11, -9, -2], [25, -7, 1, 0, -2, -5, 2, 0, -6], [8, 0, 10, 3, -1, 8, -11, -3, 6], [-7, -9, 2, -8, 4, -1, -8, 6, -6], [-8, -1, 10, 0, 2, 4, -11, -4, 5], [30, 0, -12, 6, 8, 3, 7, 3, -2], [-1, -9, 11, -7, -2, -6, -10, 13, 2], [4, -1, 7, 2, -6, -9, -4, -3, 1], [10, -7, 5, 1, 0, 0, 1, 5, -8], [18, -2, -4, -1, -2, 8, -3, -2, -2], [-6, 4, 5, -10, 2, 7, 1, 3, 8], [-24, 3, 13, -4, -13, -8, -16, -4, 0], [11, -6, 0, -1, -8, 7, -11, 1, -4], [14, 6, 6, -13, 1, -6, 5, 9, 0], [-43, -8, 14, -16, 0, 10, -22, -4, 13], [-17, -8, 3, -1, 10, -1, 8, 2, 9], [-1, -2, 5, 4, -3, 1, -6, -2, 2], [-19, 16, 4, -2, -5, -2, -13, -2, 10], [25, -7, 1, -6, 3, -1, 0, 9, -2], [7, 8, -14, 8, -5, -3, 12, -9, -4], [14, 1, 11, -10, 2, 8, -5, 2, -1], [17, -18, 8, -13, 14, 10, 4, 8, -6], [-16, 16, 4, -12, 1, -1, -14, -2, -1], [-20, -4, 4, -6, -1, 7, -5, 1, 10], [-29, 10, 9, -14, -1, 2, -8, -9, 7], [-15, -8, -1, -10, -10, -6, -5, -6, 8], [27, 8, 2, 3, -1, -5, 13, 0, -6], [25, 6, -2, 8, -4, -4, 10, 0, -7], [-14, -5, 3, -1, -12, 2, -19, -8, 8], [-12, 4, 13, -16, -5, -5, -13, 3, 0], [19, -4, -6, -4, 9, 2, 9, 1, -8], [-6, 2, 5, 6, -4, -9, 0, -1, -11], [22, 9, -4, 10, 14, -6, 11, 8, -9], [-16, 12, 2, -10, 1, 10, -7, -15, 13], [39, 3, -14, 13, 0, 4, 7, -10, -11], [5, -18, -4, 5, -1, -6, 3, 3, -7], [24, 5, -6, 7, -1, 4, -2, -9, 5], [0, -9, -1, -10, -4, -1, -2, 2, 3], [-3, -3, 7, -2, -5, 6, -6, 2, -3], [-1, 0, -5, -2, 4, -8, -7, 7, -4], [10, -18, 0, 1, -1, -12, -5, 15, -10], [6, -10, -1, 3, 2, 1, 9, 2, -7], [-7, 10, -1, 2, -14, 4, -12, -13, 11], [24, 10, -16, 17, 2, -3, 23, -7, -9], [-9, -3, 5, -12, 4, 8, -5, 8, 4], [-1, 2, -6, 5, -7, -4, 4, 6, 8], [6, -4, 8, -6, -2, 4, 0, 10, 4], [-6, -11, -7, -1, -10, -13, 0, 7, -4], [-7, 2, 0, -4, -4, -12, -5, 6, 4], [11, 6, -5, 0, -1, -9, -2, 1, 0], [18, -4, -7, 11, 6, -1, 7, 11, -1], [18, -5, 1, 1, 7, -1, 7, 8, -15], [-9, -10, 5, 9, 7, 0, -1, 12, -5], [7, 13, -4, 0, -7, -5, -2, -10, 8], [6, -4, 1, -1, 10, 15, 10, -3, 4], [-21, 0, -2, 1, 1, 3, -4, 5, 11], [29, -20, -4, -5, 2, 4, 6, 4, -6], [8, -7, 2, 2, -2, 12, -1, 2, 5], [3, 12, -8, 2, 7, -2, 17, 0, -2], [8, -3, 3, 5, 1, 12, -5, -6, 8], [11, 2, -7, -7, -2, -1, 6, 0, 5], [17, 3, -3, -7, 8, -2, -1, 7, -3], [2, -10, -5, -5, 0, 5, -10, 4, -4], [-2, 1, -3, 17, 5, 4, 4, -7, 1], [4, 23, -7, 12, -9, -1, 5, -5, 0], [-6, -7, -1, -3, 1, -3, -5, -11, 1], [-12, 6, -7, -1, -7, -4, -9, -8, 10], [-4, 1, 4, 1, -3, 0, -10, -4, 13], [12, 0, 0, 6, 2, 4, -6, -16, 2], [-3, 8, 2, 5, 13, 5, 15, 4, -12], [-7, 11, 4, 3, -2, -7, 5, 3, -2], [20, 2, -11, 13, -2, 10, 10, -10, 3], [0, 3, -2, 7, 5, 1, 11, -2, -5], [6, 7, -15, 22, 2, -6, 13, -7, -3], [-7, 1, 8, 1, 3, 3, -3, 2, 2], [-17, -12, 4, -13, -21, -2, -19, -7, 3], [-32, 23, 13, 0, -7, -10, -12, -2, 4], [-1, -5, -9, -10, 0, 7, 2, 0, 4], [-11, -3, -14, 0, 0, -5, -7, 5, 5], [-14, -11, 7, 0, 1, 10, 4, -10, 2], [-5, 16, -6, 4, -1, -9, 0, 1, 10], [10, 12, 6, 8, 19, 1, 14, 2, -7], [-2, 3, -8, 2, -5, -6, -3, -7, 3], [17, 5, -12, 6, 17, 12, 18, 1, 1], [-9, -8, 14, -3, 11, 7, -7, 6, -6], [-8, -16, 6, -1, -2, -8, -2, 15, -1], [18, -1, 2, 2, 14, 2, 13, 4, -15], [3, 9, -3, 0, 2, -7, 3, 2, -17], [-7, 6, -1, 2, -14, -8, -6, -3, 7], [-27, 1, 2, 2, -7, -11, 1, -7, 5], [20, -6, 5, 2, -6, -10, -10, -2, -4], [9, 11, 6, 6, 0, 1, 2, -5, 2], [8, 5, -12, 3, -1, -7, -2, -1, 4], [-17, -6, 16, -13, -10, 4, -20, 2, 8], [-7, -3, -12, 15, -5, 3, -5, -4, 0], [18, 16, -10, 14, 1, 1, 9, -6, 4], [-4, -12, 5, -1, 9, 3, -2, 11, -4], [1, 0, -4, -2, 5, -2, 14, -1, -7], [1, 5, -7, 4, -5, 3, 7, -5, 3], [-10, -8, -1, -8, -6, -5, -12, 6, 1], [-12, 8, 5, 8, -4, 4, -2, -15, 4], [-4, 7, 5, 2, 10, 3, -5, 4, 2], [-23, -12, 12, -17, -7, 0, -14, -3, 2], [2, 3, 15, -10, 9, -2, -2, 4, 4], [-13, -16, 16, -8, -8, -10, -6, 14, -1], [-14, -13, 3, 2, 5, 7, -5, 1, 2], [14, 4, 8, -5, -8, 0, -16, -10, 3], [12, 1, -4, -1, 11, -2, 13, 8, -6], [-16, 7, 1, -5, 4, -4, -13, 10, 4], [-4, -14, -5, 7, -11, -10, -7, 1, -12], [-43, 8, 13, -3, -5, 2, -23, 6, 3], [12, 16, -4, -3, -4, -10, -7, 6, -10], [23, 7, -7, 17, 4, -4, 13, -9, -11], [31, -9, -1, -6, 12, 11, -2, 10, 0], [5, -9, 2, -8, -6, -5, -8, 6, 0], [9, -1, 3, -3, 15, 14, -2, -6, -2], [2, 10, 4, 3, -5, -11, 10, 6, -6], [4, 3, -4, 15, 4, 1, 6, -6, 10], [-29, 12, 4, 6, -7, 9, 0, -22, 12], [24, 7, -9, -6, 5, -10, 10, 4, -4], [20, 9, -2, 6, 10, 7, 8, -1, -7], [2, -14, 9, 8, -4, -5, -3, 4, 10], [-27, -14, -4, -3, -3, -2, 1, 5, 5], [3, -7, -6, 4, 8, -4, 7, -2, -8], [9, -2, 7, 10, -5, -4, -11, -4, -2], [21, 8, -9, 7, -1, -12, 16, -4, -6], [-6, 23, 7, -1, -8, 0, -5, 0, 14], [48, 6, -21, 21, 1, -7, 24, -5, 0], [3, 16, 0, 13, -3, -6, -1, -8, -9], [36, -8, -11, 4, 6, -5, -1, 13, -4], [-40, -6, 4, 3, -17, -6, -10, -2, 7], [10, -3, -6, 2, 0, 3, 7, 3, -8], [-31, 1, 19, -11, -9, 10, -26, -1, 18], [-8, -6, -4, -12, 5, 13, 0, 0, -3], [-2, -12, 2, 0, 3, 18, 1, -11, 3], [-10, 8, 11, -13, 3, 10, 3, 3, 8], [-7, 3, -6, -1, 6, -1, 11, 7, 2], [7, -19, -10, 4, 11, 2, 10, 6, -11], [-45, 19, 8, -4, -5, 6, -11, -10, 10], [-24, -8, -4, -7, 2, -8, 5, 14, -10], [8, -5, 2, -4, 9, 7, -13, -2, -6], [28, 13, -5, 23, 1, -3, 19, -3, -11], [11, 3, 5, -7, -10, 0, -16, -1, 2], [8, 1, -7, -5, 8, 3, 4, 7, 0], [21, -2, -14, 8, -3, -5, 24, 4, 0], [-6, 5, -8, -10, -8, -4, -5, -4, 13], [8, -15, 14, -13, -3, 5, -14, 13, 4], [10, 6, -12, 4, 2, 3, 15, 0, 4], [-25, 2, 10, 10, 3, -4, 3, -6, -2], [11, -10, 9, -10, 0, 8, -10, 5, 1], [-14, 5, -2, 17, 3, 2, 6, -1, -7], [-1, 6, -14, 4, 6, -11, 13, 3, -11], [-18, -5, 2, -1, 5, -7, -1, 7, -11], [1, 9, -15, 2, 4, -6, 13, -3, 0], [13, -8, 3, -4, -13, 1, -11, 21, 1], [-25, 13, 0, -7, -2, 6, 4, -17, 2], [-29, -9, 7, -12, 0, -6, -18, -1, -7], [10, 11, 2, 0, 10, -1, -6, -13, -7], [-30, 15, -2, 5, -3, 0, -6, -13, 15], [3, 9, 11, 2, 3, 3, 11, 5, -11], [22, 8, 6, 5, -7, 9, -7, -15, 7], [11, 23, -5, 15, 0, 6, 12, -11, 0], [-19, 7, -12, -1, -4, 6, -1, -4, -5], [17, 0, 10, -2, 16, 4, 2, 5, -9], [-14, -20, 7, -18, -1, -5, -12, 1, 1], [-27, 3, -5, 1, 9, -12, 10, 14, -2], [-10, -15, -5, 10, -5, 1, 8, 0, 3], [10, 11, 0, -5, 9, -5, 1, 5, 1], [-11, 6, 8, -20, -8, 2, -22, 5, -1], [-3, 5, 9, 6, -6, -10, 5, 1, -8], [31, 1, -17, 17, 17, -8, 30, 0, -10], [-39, 4, 12, -5, -16, -1, -6, -11, 7], [-6, 13, 2, -3, -1, 6, -6, -10, 17], [-13, 14, 0, 8, -7, 10, -1, -11, 6], [37, -8, -3, 4, -3, -8, 9, -6, 0], [-32, 10, 10, -12, 2, 4, 0, -5, -2], [-28, 20, -9, 8, -13, -6, -4, -21, 2], [-10, 5, 2, 1, 3, -4, -8, 11, 1], [-44, -2, 11, -3, 3, 5, -5, 3, -3], [23, -1, -15, 0, 14, -2, -3, 8, 0], [-13, -13, -6, 2, 10, 6, 1, -6, 4], [3, 10, -6, -6, 11, 8, 7, -9, 2], [33, -5, -8, 2, -5, -6, 6, 10, -9], [14, -16, 14, 1, -7, -6, 3, 9, -2], [40, 10, -4, 9, 12, -8, 24, 10, -7], [-27, -4, 1, -11, -7, 4, 1, 2, -3], [-4, 6, 4, 4, 0, -10, -2, 6, -4], [-3, 10, -2, 7, -14, -10, -16, 6, 4]];
    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_f();".
// 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_f(: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_f();".
// 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_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8004_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-24, 60, 26, -118, -1, 75, -4, -17, 1, 1]>,<7,R![552, -212, -834, 146, 463, 7, -101, -16, 5, 1]>],Snew);
    return Vf;
end function;