
// Make newform 4004.2.a.i in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4004_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_4004_2_a_i();".
// 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_4004_2_a_i();".
// 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 := [-87, -67, 271, 74, -233, 15, 60, -12, -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], [0, 1, 0, 0, 0, 0, 0, 0, 0], [-5712, 5318, 17667, -8299, -6837, 2658, 627, -189, -10], [-1623, -5365, 2817, 5000, -2712, -1257, 705, 84, -46], [-1101, 5365, -2136, -5000, 2712, 1257, -705, -84, 46], [-11703, -6449, 20334, 7126, -9129, -1734, 1491, 111, -77], [6417, 6374, -15249, -14926, 11235, 4839, -2727, -351, 176], [9774, 18175, -26214, -29312, 19464, 8913, -4656, -633, 298], [-7044, -8215, 16717, 12553, -10533, -3804, 2315, 271, -143]];
    Rf_basisdens := [1, 1, 681, 681, 681, 681, 681, 681, 227];
    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_4004_a();"
function MakeCharacter_4004_a()
    N := 4004;
    order := 1;
    char_gens := [2003, 3433, 365, 925];
    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_4004_a_Hecke(Kf)
    return MakeCharacter_4004_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, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 0, -1, 0, 0, 0, -1, 0], [1, 0, 0, 1, 1, -1, 0, 0, 0], [-1, 0, 0, 0, -1, 0, -1, 0, 0], [0, 0, 0, 0, -1, 1, -1, 0, 0], [1, -1, 1, 0, 0, 0, -1, 0, 0], [1, 1, 0, -2, -1, 1, 0, -1, -2], [2, 1, -1, 0, -2, 1, 1, 0, 0], [1, 0, 1, 1, 1, -1, 0, 0, -1], [2, -1, 0, -1, -1, 2, -2, 1, -1], [-1, -1, 0, -1, -1, -1, -1, -1, 0], [-1, 1, -1, 0, -1, -1, 0, 0, 0], [5, 0, 0, -2, -1, 2, 0, 0, -1], [2, -1, 0, 2, -2, 1, 0, 1, -1], [0, 2, 0, -3, -2, 1, 1, -2, 0], [2, 0, -1, 0, -1, 1, -1, -1, -2], [1, -3, 1, 0, 0, -2, -1, -1, 0], [1, 1, -1, 0, 1, 0, -1, 2, 1], [0, 0, 0, 0, -1, 0, -2, 1, -2], [3, 0, 0, -2, 0, -1, 2, -2, 0], [-1, 1, 1, 2, 1, 0, 1, 0, 3], [-4, 3, 0, 4, 2, -1, 1, 3, 0], [1, 1, -1, -4, -2, 1, 0, -1, 0], [3, -2, 0, 1, 2, -1, -2, 1, 1], [0, 1, 1, 0, 0, -2, 0, 0, -1], [1, 2, 1, -3, 0, -1, 0, 0, 3], [2, -1, 1, 0, 2, 1, 1, 2, 0], [-1, -2, 1, 1, 3, -2, 1, -1, 1], [6, -4, 0, 1, 1, -1, -1, 0, 1], [1, 3, -1, 1, -1, 3, 2, -1, -1], [-6, 4, 0, -1, 0, -2, 0, -2, -1], [1, 3, -1, -3, -2, 0, 3, -2, -1], [1, 1, 1, -2, -2, 3, 0, 2, -1], [-1, -4, 2, -2, -2, 1, -1, 0, -3], [-3, 0, 1, -1, -3, -1, 2, -1, 0], [3, -2, 1, -2, 3, 0, -1, 1, 0], [5, -6, 1, 3, 1, 0, -1, 5, 1], [-9, 3, -1, -1, -3, 1, 0, -2, -1], [-1, 0, 0, 4, -1, 1, 2, 1, 0], [5, 3, 0, 0, 2, -1, 0, 2, 0], [9, -2, -1, 0, -3, 4, 1, 0, -2], [-1, -3, -2, 5, 1, -2, -1, 2, -1], [7, -2, 2, 1, 3, 0, 1, 2, -1], [1, 3, 1, 0, -1, -1, 1, -2, -5], [8, -3, 0, 4, 3, 0, 2, 4, 2], [10, -1, -1, -1, 1, 0, 2, -2, -3], [4, -1, 0, 5, 3, 0, 0, 1, 3], [6, -6, 1, 3, 2, 0, 0, 3, 1], [8, -5, -1, -1, -2, -1, -3, 1, -3], [-2, 1, -2, -2, 0, 1, -1, 1, 0], [-5, -4, -1, 0, -2, 3, 0, -2, -2], [4, 0, -2, 1, 1, -1, 5, -2, -1], [1, 1, 1, 3, 2, 0, 0, 0, -4], [8, -1, -3, -3, -4, 3, 3, -2, 1], [1, 4, -2, 1, 1, -1, 0, 1, -1], [8, -1, 2, -1, 0, 3, -3, -1, -1], [6, -6, 1, -1, 0, 0, -2, -1, 3], [-1, 1, 0, 3, 3, -3, 3, 1, 2], [-3, -2, 2, 1, 2, -3, 0, 5, 3], [-1, 2, -1, -4, -6, 1, 2, -1, -4], [5, -2, 1, -4, -2, -1, 0, -1, -2], [-6, -1, 2, 2, 3, -3, -3, -3, 1], [4, -1, 2, -2, -4, 1, -3, 1, -2], [-9, 4, -3, -5, -4, 1, 2, -4, -1], [-8, 4, 0, -5, -4, -2, -2, -4, -1], [-5, 5, 0, -7, 2, -4, 1, -3, 3], [-2, 3, -1, 3, 4, 1, 1, 2, 5], [-6, 0, -2, 0, 0, 1, 3, -3, 2], [-4, -2, 3, 4, 3, -3, -1, 3, 4], [-2, 6, 0, -7, -2, 2, 1, -2, -2], [7, -7, 1, -1, 0, 2, -1, 0, 3], [-8, 5, -1, -1, -2, 2, 4, -2, -2], [5, -2, 1, -7, 1, 1, -2, 2, 3], [8, -4, 0, 4, 3, -1, 3, 2, 4], [0, -3, 3, 2, 1, -4, -2, 0, 2], [-7, -4, 2, 3, 0, 4, 1, 3, -1], [10, -3, 2, 3, 8, -1, 3, 3, 7], [0, -4, -4, -2, -1, 1, 1, -6, -2], [-9, 3, 0, -5, -5, -1, 0, -7, -5], [4, -4, 0, -1, -2, 2, -4, 2, -3], [1, 2, -1, 5, 3, -2, 1, 1, 3], [5, 3, 1, 5, 0, 0, 1, 4, 1], [5, 5, 1, 3, -1, -1, 0, 3, -1], [4, 2, -2, -3, -3, 3, 3, -6, -1], [-7, 7, -3, 2, 4, -2, 3, 0, 6], [10, -6, -3, 3, 2, 5, -3, 6, 1], [-7, 0, 2, 0, 5, -5, -2, 0, 1], [-7, 1, 2, 1, -1, -3, 2, 1, -1], [4, 3, 0, 0, -3, -2, -2, -1, -2], [-3, -2, -2, 2, -1, 3, -4, 2, -1], [-2, 3, -2, 1, 2, 1, 3, -3, -1], [3, 4, 3, 0, 0, 1, 2, -1, -4], [-5, -7, 1, 1, -4, 0, 1, -2, -3], [-6, -3, -1, 7, 1, 0, -2, -2, -5], [6, -3, 1, -5, -2, -1, 1, 0, -5], [-10, 9, 0, 1, 0, 1, 1, -3, -1], [-9, 5, 2, -3, 5, -3, -2, -2, 3], [8, -5, 2, -1, 2, -1, -3, 5, 9], [4, 3, -2, -3, 0, 3, -1, -1, -3], [-9, -5, 2, -4, -4, 2, 1, -2, -4], [4, -2, 0, 0, -2, 5, 3, 3, -4], [4, -7, -1, 7, 2, 3, -3, 4, 1], [-14, 0, 1, 6, 3, -5, -1, 5, 4], [-8, -4, -3, 2, 0, 2, 2, -1, -2], [9, -3, 1, 0, 3, -1, 3, -6, 3], [-6, 5, 0, -6, -1, 0, 0, -7, -6], [6, -3, -1, -4, -3, 2, -6, 2, -4], [-10, 8, 2, -1, 4, -6, -4, -4, 5], [9, 2, -1, -6, -1, 0, -3, -3, -2], [-2, -6, 2, -1, 5, -3, 2, 0, 4], [-4, -1, 1, 10, 7, -4, 3, 6, 9], [-3, 3, 0, 2, 1, 1, 0, 3, 6], [-1, 7, -3, 4, 1, 1, 2, 2, 0], [0, -10, 0, 9, 5, 1, -4, 8, -2], [1, -6, 2, -3, 0, 4, 1, -1, -1], [8, -2, 0, 2, 0, 0, 2, -1, 4], [14, 10, 0, -5, -1, 1, 1, 4, 1], [5, -7, 0, 2, 5, -1, -3, 3, -5], [-5, -2, 3, 3, 6, -1, 5, 3, 2], [3, -3, -2, 4, 4, -2, -1, -3, 4], [0, 3, -1, 4, 4, 3, 3, 2, -2], [-3, 2, 1, -3, 0, 1, 6, 1, 3], [-8, -2, 0, -1, 4, 1, 5, 1, 7], [-10, 2, 0, -1, 4, -4, 2, 1, -1], [6, 7, -4, 0, -4, 5, -1, 3, 0], [4, -4, -1, 5, 1, -1, 5, 7, 1], [-20, 8, 0, 0, 1, -5, 5, -10, 0], [-1, 4, -4, 3, -1, 0, -3, -2, 3], [13, -10, -1, 4, 1, -1, -4, 3, 1], [-6, 2, -1, 3, -3, 1, 3, 1, -3], [22, 3, -2, -9, -8, 1, -5, 3, -1], [9, 0, -3, -1, 1, 3, -8, 8, 1], [7, -1, -2, 5, -8, 4, 2, 5, -2], [9, 1, 1, -4, 0, 0, 1, 0, -4], [-6, 2, 0, -3, 2, 4, -4, -4, -5], [1, 4, 0, -1, -1, -5, 2, -6, 0], [3, 2, 5, -2, 7, -3, 0, 2, 8], [-1, -4, -1, -5, -2, 3, -2, -5, -3], [-4, 5, 1, -7, -4, 5, 1, 2, 1], [-7, -2, 3, 2, 3, -2, -1, 3, 2], [-7, -3, -2, 3, -1, -3, -2, 3, 1], [6, -4, -1, 2, 3, 3, 3, 7, 4], [13, -8, 1, -7, -2, 4, -3, 3, 4], [1, -7, 3, 2, 0, 0, -3, -2, -4], [-22, -6, 3, 7, 5, -5, -7, -1, 1], [3, -3, 1, 4, -1, 2, 3, -5, 0], [7, 4, -1, 1, -3, 0, 1, -5, -3], [-6, -8, 0, 2, -3, 1, 3, 0, -4], [-14, -5, 3, -6, -4, -1, -3, -6, -6], [-2, -2, 3, 0, -2, 0, -4, 5, -4], [6, -3, 1, -8, -4, 3, 1, 0, 6], [-7, -3, -2, -2, -7, 8, -3, -2, -2], [-13, -1, 2, 1, 0, -3, 0, -2, -3], [-5, 7, 1, 7, 0, -1, 4, 4, 2], [0, -5, 0, -4, -8, 3, 1, 5, -6], [-10, 3, 1, -10, -4, 1, -2, 4, -1], [18, 0, -3, -3, -7, 3, 1, 1, 1], [0, 2, -3, -2, -3, -1, -1, -1, -2], [-10, 2, 1, 1, -6, 0, -4, 2, -9], [-5, 6, 1, 1, 3, 2, 1, -5, 1], [-23, 2, 2, -5, -2, 3, -2, -2, -1], [-22, 4, -4, 2, -2, 4, -2, 8, -4], [-1, 0, -2, 1, 0, 0, -1, -4, -8], [1, 4, -1, 2, 7, 0, -1, 2, 6], [-21, 2, -1, 10, 8, -7, 6, -3, 4], [2, 4, -1, 2, -9, 5, 3, 3, -6], [28, 0, 0, 2, 1, -1, 5, 2, 2], [-3, -1, 2, 5, 6, -2, -1, 10, -1], [7, -2, 0, 12, 8, 0, -1, 3, 2], [-4, -9, 5, 2, 7, -3, 7, 3, 2], [-17, 17, -6, -8, -4, 2, 7, -11, 0], [6, -6, -3, 2, -1, -3, -5, -5, 0], [-11, -4, -5, 1, -1, 0, -1, 3, 1], [-2, -3, 2, 8, -1, 2, -2, 5, 2], [7, 3, -3, -11, -12, 9, 2, -8, -6], [-7, 10, -4, -2, -4, 2, 5, 0, -5], [6, -3, -2, -8, -7, 4, 4, -3, -2], [-18, 4, 1, -4, 0, -5, 1, 1, -1], [-14, 4, 2, 8, 6, 0, 0, 6, 2], [-14, 1, -3, 4, -1, 2, -1, 2, -7], [8, -1, -1, 3, -2, -1, 5, 4, 1], [-20, -7, -1, -5, -2, -5, -1, -4, -3], [-8, -5, 2, -7, 0, 1, -3, -7, -7], [13, 3, 0, 7, 1, -1, 4, 2, 7], [20, 2, -1, -12, -4, 8, 2, 3, -2], [-23, 6, 4, 5, 7, -8, -5, 6, 5], [8, 4, -2, -4, -3, 1, -1, 2, -4], [-7, 5, -6, -3, -1, 1, 10, -7, 3], [17, -5, -5, 5, 1, 0, -7, 3, 1], [0, -6, -2, -2, -3, 3, 1, -2, -6], [-9, 3, 1, 2, 4, 4, 7, 2, -2], [3, 3, -1, -13, -1, -2, 1, 1, 5], [8, 1, 4, -3, 5, -2, 1, -7, 0], [-8, 12, 3, -5, 4, -2, 4, -1, 1], [0, -6, -5, 8, 3, 3, -7, 9, -2], [5, 5, -1, -2, -4, 3, 2, 3, 8], [-10, -3, 2, 2, -1, -2, -4, -5, 4], [5, -1, 1, 0, -3, -3, -6, 3, 0], [-1, -5, -4, 4, 1, 4, -7, 9, 3], [-3, -3, 2, 13, 8, -4, 3, 13, 9], [-7, -4, 2, 13, 4, 3, 2, 0, -1], [1, 15, -3, -4, -7, 3, 0, -8, -2], [-4, 7, 0, -8, 2, -1, -3, 3, -4], [23, 8, -1, -4, -1, 4, -1, -6, -6], [23, -7, -4, -1, -7, 3, -4, 0, 1], [4, 6, 0, -5, -1, -1, 3, -2, 3], [-3, 9, -4, -3, -2, 0, 5, -9, -3], [-5, -19, 5, 9, 1, -1, -2, 6, 3], [18, 4, 1, -3, 6, 0, -2, 1, 5], [13, -5, -2, 2, -2, 3, 2, 9, 5], [-8, 6, 0, -8, -9, 5, 7, -7, -2], [1, -2, 3, 2, 10, -8, 3, -2, 4], [10, -12, 3, 13, 10, 0, 2, 13, 1], [-11, -2, 0, 3, 1, -2, 0, 6, 8], [-15, 2, 1, -15, -5, 2, -1, -5, 1], [-6, 2, 7, 1, 7, -9, 5, -1, 1], [-8, -3, 2, 6, -4, -1, 1, 1, -4], [17, -12, -1, -5, -4, 3, 0, 3, -7], [-4, -1, -6, -3, -6, 2, 2, 1, -4], [-4, 7, 4, -11, -2, -1, 1, -3, 5], [2, 7, 4, -4, 1, -2, 0, -1, 2], [-20, 10, 0, 6, 2, 0, 2, 0, 6], [-2, 7, 1, 3, 14, -7, 1, 2, 9], [2, -2, -1, -5, -8, 0, 0, -7, 3], [2, -4, 0, 1, 4, 2, 8, 12, 3], [-2, 7, 0, -1, 3, 0, -5, 3, -2], [-6, 0, 2, -2, -1, 3, -5, 4, 4], [10, -7, 4, 7, 10, -7, -5, 5, 7], [-12, -10, 3, 3, 0, 0, -10, -1, -3], [18, 10, 1, 3, 5, -1, -3, 2, -1], [19, 15, 0, 3, 5, 1, -1, 7, 2], [9, 0, -3, 1, -1, 6, -5, 1, -3], [5, -4, 2, -4, 0, 3, 2, 3, -4], [-5, 2, -4, -1, -4, 5, 2, 10, -1], [6, -12, -4, 1, 1, 0, -2, 1, -1], [7, -6, -3, -6, 0, -3, 2, -3, 4], [30, 3, 2, 3, 5, -4, 6, 5, 5], [15, 7, 2, 1, 1, 4, 1, 0, 1], [29, -3, -2, 5, 1, 1, 2, 5, 11], [12, 8, 1, -11, -8, 2, 6, -5, -7], [-2, 7, 5, -4, 7, -4, 6, 2, -2], [8, 9, -6, -8, 0, 5, 3, 5, 2], [4, 0, 5, -6, -8, -4, 2, -5, -2], [-8, -11, 8, 12, 13, -6, -4, 1, 0], [4, 1, -1, 5, -7, -2, 0, 2, 3], [23, -8, 1, -4, -9, 12, -5, 2, -10], [-2, 0, -5, 5, 1, 1, -1, -3, 7], [4, -1, 3, 9, 7, 2, 3, 6, -4], [-8, 5, 2, -4, -1, -4, -6, -5, 2], [-11, -3, 0, -1, -6, -1, 2, -7, -12], [-16, 3, 3, -9, -7, -4, -2, -2, -5], [-11, 2, 3, 0, 6, -1, 5, 5, -3], [15, 6, 0, 0, 0, -6, 9, -8, 11], [11, 1, 4, 1, 1, -5, -2, -13, 1], [22, -13, 0, 1, 3, 0, -8, 9, -3], [-3, -4, 1, -3, -4, 8, -3, 10, 1], [5, 3, -5, -9, -4, 9, 4, -3, 1], [2, -5, -2, 6, -2, -3, -1, -7, -6], [19, 2, -6, -8, -16, 10, -3, 2, -3], [11, 1, 4, -8, -11, 4, -5, -2, -8], [1, 6, -2, -11, -8, 2, -1, 1, 1], [-11, 0, 6, -3, 1, 2, -5, 2, -1], [6, -5, 1, 5, 4, -4, -10, 3, -1], [4, 11, 1, -5, -7, 6, -2, 2, 5], [-24, 5, -4, -2, -3, 4, -6, 9, 0], [11, 2, -3, 1, 1, 2, -3, 3, 9], [4, 11, -1, 0, -5, 5, 5, 4, 1], [5, 3, 1, 3, -9, -3, 0, -1, -1], [-1, -3, 5, 2, 2, -1, 6, -7, -6], [-9, 7, 3, 1, 9, -11, -7, 2, 4], [3, -6, -2, 6, 10, 1, 2, 0, 10], [-8, 3, -1, -1, 0, -2, -8, -4, 8], [9, -4, 0, 2, 5, 7, 4, -5, 2], [22, -13, 2, 7, 3, -8, 2, 1, 3], [14, 0, 5, 5, 6, -2, -4, 7, -1], [-12, 7, -4, -4, -2, 1, -3, 5, 4], [-10, 12, -5, -9, -2, 5, 3, -1, -2], [-8, 10, 3, 0, 1, -1, -3, -5, 0], [40, 8, 3, 3, -3, -5, 3, 3, -3], [9, 3, 0, -14, -14, 4, -4, -9, -1], [-13, -11, 3, 4, 2, -2, 1, -4, -4], [3, -1, 0, -4, -1, 0, 3, -3, 9], [9, -2, 0, 5, 1, 1, 6, 5, 3], [-17, -1, 6, -2, -3, -9, -10, -3, 4], [23, -13, 4, -7, 2, 0, -2, 5, 2], [9, -1, -3, 0, -8, 0, -5, -4, -4], [-18, -7, 10, 3, 8, -7, -7, -1, 1], [-11, 8, 4, 4, 13, -6, 9, 9, 8], [1, -12, 9, 5, 3, -9, -8, 9, 6], [-4, -11, 0, 4, 12, -3, 5, 3, 10], [4, 12, -6, 7, -4, 2, 0, 6, 1], [-10, 6, 2, 14, 5, 0, 2, 11, 6], [-7, 2, -3, 7, 8, -1, 4, -11, 5], [-4, 0, 1, -11, 6, -6, 2, -3, 11], [26, 9, -4, -13, -6, 3, -1, -9, -9], [0, 2, 8, 5, 4, -8, -2, 2, 3], [-5, 2, 1, 9, 5, -3, 0, 16, 5], [8, 12, -7, -1, -6, -4, 2, 1, 5], [-26, 17, 0, 14, 1, -6, 2, -3, -4], [23, 14, -6, 7, -3, -2, 9, 0, 11], [11, 14, -2, -13, -3, 4, 7, 4, -1], [-1, -16, 1, 1, 3, -8, -1, -7, -5], [31, -3, 3, -7, -9, 5, -12, 8, -3], [-23, 3, 5, -4, 5, -2, 5, 4, -3], [10, -3, 1, -7, -4, 5, 5, -18, 1], [-22, 1, 0, 16, -2, 3, -1, 13, -4], [18, 3, 5, -3, 1, 0, -4, -4, 3], [24, -3, 1, -4, -5, 0, 0, 2, 8], [-23, 3, 2, 8, 5, -3, 6, 1, 8], [22, 14, -1, -6, -4, 2, -2, -7, -8], [11, 6, -3, 2, 4, -2, 11, 7, 3], [11, 13, -2, -7, -5, -1, 2, -7, 3], [24, -12, -7, -1, -3, -1, -7, -3, 5], [21, -1, 4, 1, 9, -3, 6, -2, 11], [20, -6, -1, 7, 0, 0, -4, -3, 1], [-6, 9, -6, -3, -6, 5, 3, -5, 1], [1, 10, -6, -3, 3, 2, 5, -6, 7], [11, -2, 3, -5, -6, -1, -9, 3, 8], [0, 0, 1, -1, 4, -2, -1, 7, 4], [-27, -3, -1, 7, -2, -5, -4, 1, -5], [11, 10, -3, -5, 5, -4, 3, 3, 1], [-4, -15, 2, 7, 7, -6, 1, 1, 8], [25, 6, 3, -7, -4, 3, -4, -1, 3], [17, 1, -3, 5, -5, 3, 8, -3, -1], [-4, -8, 8, 11, 8, -10, 0, -4, 3], [31, -4, 0, -1, -5, -7, -10, 1, -3], [15, -4, -5, 7, 0, 0, -3, 5, 6], [12, -1, -7, 3, -3, 6, 6, 0, -5], [1, 1, 1, 9, 8, -9, -4, 0, 2], [2, 0, 1, 2, -10, 0, 4, -5, -6], [-5, 11, 5, 4, 8, -2, -3, 0, 8], [11, 11, -8, 1, -1, 5, 4, -7, -3], [-4, 7, -5, 8, -1, 2, 0, 6, -2], [-8, -4, 3, -1, 0, 4, -8, -4, 1], [-4, 11, 4, 2, 2, -5, -7, 7, -6], [-11, -3, 8, 2, 1, -5, 4, -13, -8], [-23, 9, 3, 0, 1, -1, 4, -8, 2], [-30, 0, 5, 5, 9, -3, -9, -5, 1], [11, 13, 1, -2, 9, -7, -8, 6, 4], [3, 2, -6, 4, -9, 2, -7, 6, 2], [15, 5, -7, -7, -3, 1, 2, -3, -7], [-3, -5, 2, 3, 4, 8, 11, 5, -3], [-9, 7, -4, 1, 4, 2, 3, -3, 1], [0, 20, -5, -2, -2, 8, 14, 1, 4], [19, 4, 3, -17, -9, 6, -3, 1, -5], [23, -1, 1, 6, -3, 3, 5, -10, -1], [3, 0, -1, -13, -12, 11, -4, -11, -5], [20, 17, 5, 1, 2, 3, 1, 6, 1], [-6, 14, 3, 3, -2, -8, 8, -5, 3], [0, -1, 3, 2, 2, -9, -9, -2, 6], [-14, -9, 7, 24, 17, -8, -2, 6, 6], [31, -3, -6, 8, 4, 0, 7, -3, 0], [36, 5, 1, 1, 2, -5, 3, 2, -5], [-46, 8, 0, -6, -1, -5, 7, -8, -2], [10, -10, -4, -6, -10, 11, 5, -6, -1], [-7, 2, -6, -13, -10, 12, -1, -3, -9], [-13, -10, 0, -5, 5, 1, 0, 2, 6], [29, -1, 4, -5, -3, 9, -6, -3, -5], [2, -13, 0, -13, -12, 5, -3, -9, -5], [22, -5, 2, 3, 0, -8, -8, 4, -3], [-8, -6, 6, -8, -9, 0, -10, 0, -9], [28, 9, -7, 7, 3, 6, 0, 0, 3], [-20, 5, -2, -5, -5, 0, 2, -3, -9], [0, -20, -2, 14, 1, 1, -9, 8, -2], [3, 5, -7, -5, -5, 5, -6, 2, -5], [9, 2, 6, -7, -8, -7, 2, -9, 1], [4, -5, -3, -5, -1, -2, 6, -10, 5], [16, 3, -1, 1, 4, 5, 13, 4, -3], [-16, -14, -1, -1, 8, -10, 2, -9, 1], [4, -3, 4, 7, 7, -12, -5, -7, -8], [8, -5, -5, -8, -17, 8, 6, -4, 4], [-1, -1, 4, -1, 3, -1, 6, 6, 9], [-6, -6, -3, -2, -7, 15, 5, 7, -2], [-9, 1, 9, 2, 4, -2, 0, 6, 9], [1, 0, 0, 14, 3, -9, 6, -6, 7], [11, 4, -8, -6, -7, 0, 11, -5, 10], [-1, -4, 7, 2, -2, -5, -6, -7, 2], [-38, 3, 3, 19, 9, -8, -4, 10, 7], [-3, -8, 1, 8, 9, 6, 9, 12, 4], [22, -3, 1, -15, -1, -6, 2, -18, 5], [3, -10, -4, 5, -8, -1, 8, 6, 9], [20, -12, 2, 5, 1, 1, -5, 6, 9], [3, 9, -4, -2, -5, 11, 6, 7, 2], [-20, 3, -3, 4, -6, 3, 3, -2, 10], [-16, 8, 1, -4, 6, 2, 2, 2, 14], [-5, -1, 3, 16, 2, 2, 1, 12, 2], [1, -6, -1, -8, 3, -4, 1, -19, -6], [-30, 12, -4, -2, -5, 1, 4, -8, -3], [6, 10, -2, 3, -8, 8, 10, -1, -9], [4, -3, 2, -8, -9, 2, 5, -15, -11], [-16, 1, 2, -4, -8, 1, 9, -5, -8], [-17, 16, 3, -6, 2, -4, -9, -3, -1], [-10, 15, -9, -10, -4, 5, -3, -2, 0], [0, 5, -2, -8, -15, 10, -2, -5, -8], [7, -15, 1, 0, -6, 8, 1, -3, -8], [18, -9, 5, -4, 7, -2, 4, 10, -4], [-58, 0, 4, 10, 10, -2, 5, -2, 3], [14, 1, -1, -6, -4, 15, 1, -8, -8], [-11, 3, -4, 5, -3, 0, 7, -4, 7], [-16, -1, 4, 0, 11, -2, 6, -1, 2], [16, -12, 2, -12, 1, -1, -1, 2, 0], [12, 6, 2, -6, 2, -6, 0, 8, 12], [37, -7, -9, -7, -9, 13, 8, -5, -9], [6, -8, -6, -14, 3, -1, -3, -10, 0], [15, -1, 3, 0, 5, -9, 8, 4, 0], [-17, 5, -3, 7, 13, 4, -1, -2, 2], [-8, -6, 3, 11, 4, 4, 2, 7, -7], [0, 7, -2, -17, -2, 1, -3, -17, -7], [23, 15, 1, 2, 5, 5, 8, 9, 6], [24, -15, -4, -2, -13, 10, -4, -3, 0], [-21, -2, 8, 13, 15, -10, -9, 0, 5], [-8, 5, -2, -5, -7, 0, 4, -11, -3], [-26, 7, 4, 2, 0, -7, -1, 0, 0], [-23, 16, -2, -8, -4, -7, 4, -11, -6], [0, -5, -3, -8, 0, -1, -1, 4, 0], [6, 10, -5, -11, -10, 3, 1, 1, 2], [16, -7, 3, 8, 9, -4, -8, 4, 2], [15, 5, 3, -11, 2, -1, 4, -10, 10], [18, 9, -2, -3, 7, -6, -4, -7, 1], [-1, 0, 5, -13, 0, -4, 5, 12, -1], [-28, -4, 9, -2, 0, -4, -4, 4, -6], [-3, -6, 1, 4, 10, -1, -12, 3, 16], [1, -9, 8, 2, 7, 7, 2, -2, -6], [13, -2, -8, -8, -5, 2, 1, -2, 2], [18, -20, 0, 15, 10, -4, 12, 2, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4004_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4004_2_a_i();".
// 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_4004_2_a_i(:prec:=9)
    chi := MakeCharacter_4004_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_4004_2_a_i();".
// 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_4004_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4004_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-87, -67, 271, 74, -233, 15, 60, -12, -4, 1]>],Snew);
    return Vf;
end function;