
// Make newform 8001.2.a.o in Magma, downloaded from the LMFDB on 06 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_8001_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_8001_2_a_o();".
// 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_8001_2_a_o();".
// 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 := [-8, -12, 116, 146, -372, -211, 467, 61, -242, 19, 53, -10, -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, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [16, -12, -334, -190, 615, 285, -411, -144, 125, 29, -18, -2, 1], [10, -10, -150, -109, 287, 198, -224, -114, 84, 26, -15, -2, 1], [48, -28, -502, 118, 871, -203, -571, 120, 169, -27, -22, 2, 1], [8, 72, -154, -586, 363, 863, -387, -470, 187, 105, -40, -8, 3], [12, 92, -154, -590, 363, 863, -387, -470, 187, 105, -40, -8, 3], [28, -44, -302, 246, 455, -387, -193, 220, 9, -51, 8, 4, -1], [-16, -30, 138, 298, -265, -498, 269, 285, -128, -65, 27, 5, -2], [48, 60, -518, -670, 883, 975, -655, -510, 243, 109, -44, -8, 3], [12, 42, -144, -478, 223, 758, -213, -419, 108, 93, -25, -7, 2], [18, 12, -276, -405, 550, 696, -493, -399, 212, 91, -42, -7, 3]];
    Rf_basisdens := [1, 1, 1, 4, 2, 4, 4, 4, 2, 2, 4, 2, 2];
    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_8001_a();"
function MakeCharacter_8001_a()
    N := 8001;
    order := 1;
    char_gens := [3557, 1144, 7750];
    v := [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_8001_a_Hecke( : Kf := Kf)
    return MakeCharacter_8001_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]];
    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, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [2, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, -1], [-1, 0, 0, 0, -1, 1, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 1], [0, 3, -1, 1, 0, 1, 1, 0, -1, -1, -2, 2, 0], [-2, 1, 0, 0, -1, 0, 1, 0, 0, 0, 1, -1, 0], [-1, -1, 1, 0, 0, -1, -1, 1, 0, 1, 0, 0, 0], [1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1], [-1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, -1, -2, 0, 0, 0, -1, 0, 0, -1], [-2, 1, -1, 1, 1, -1, 0, -1, 0, -2, -1, 0, -1], [-3, -1, 0, -2, 1, -1, 0, -1, 1, 1, 2, -1, 0], [-3, 1, -2, -1, -1, 0, 1, 0, -1, 0, -3, 1, 1], [3, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, -1], [-2, -1, 0, -1, 0, 1, -2, 0, -1, 1, 0, -1, 2], [-1, 1, 0, 1, 0, -1, 0, -1, -2, -1, -1, 1, 2], [1, 0, 1, -2, -2, 1, 0, 1, 1, 1, 1, -1, 0], [-4, 2, -3, -1, -1, 0, 1, 1, 0, -1, 0, 0, 1], [-2, -4, 2, -1, 1, -1, -2, 1, 1, 0, 0, -2, 1], [-5, -2, 0, -2, 0, -2, -1, 0, 2, 0, 2, -2, 0], [-1, 0, -1, -2, 1, -1, -1, 2, 1, 0, 1, -2, 1], [0, -3, 1, 3, 0, 1, -2, 0, -1, 0, -3, 0, -2], [5, 3, -1, 5, 0, 2, 1, -2, -3, -2, -4, 4, -2], [-1, 1, 0, 0, 0, 0, 0, -3, -2, -1, -1, 1, 0], [1, 0, 0, -1, 0, 0, 0, -1, 3, 1, 0, 0, -2], [-3, -1, 1, -1, -1, 1, 1, 2, 0, 1, 1, -4, 1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 1, 1, -1, 0, 1, -1, 3, 0, 0, -1, 0, -1], [-7, 2, -2, -2, -1, -2, 0, 2, 1, 0, 1, 0, 1], [-1, 5, -2, 1, -1, -1, 0, 3, 0, 0, -2, 0, 1], [-1, 2, 1, 1, 3, 1, 0, 0, -2, -2, 0, 1, -1], [0, -3, 1, 0, 1, -1, -1, -3, -1, 0, 1, -2, -1], [-1, 0, 1, 0, -1, -1, -2, 2, 2, 1, 3, -3, 1], [-4, -4, 0, -2, -1, 0, 0, 3, 1, 0, 1, -1, -1], [-5, -2, 3, -1, 2, -3, -1, 0, 1, -1, 1, 1, 0], [-5, 5, -2, 0, 2, 0, 3, -3, -1, 0, -4, 3, 0], [-2, 6, 0, 6, 2, 0, -2, 0, -4, -2, -4, 4, -2], [4, 3, -1, -1, 2, 4, 2, 1, 1, 0, -3, 1, 0], [-1, -6, 3, -5, 0, 0, -4, 2, 4, 4, 5, -3, 0], [1, -1, 1, -1, -1, -1, 2, 0, 0, 1, -2, 0, -1], [-6, -1, -1, 1, 1, -3, 0, 1, 1, -1, 1, -4, -3], [-3, 0, 0, 3, -3, -1, -3, 2, 2, -1, 0, -2, -1], [-2, -1, 0, 0, 3, 1, 0, 1, 1, 0, 1, -1, 1], [8, 4, -1, 2, 0, 2, 3, -3, -2, -3, 1, 2, 2], [-1, 4, -5, 1, -3, -2, 0, 0, -1, -2, -3, -1, 1], [-2, -6, -1, -4, -4, -3, -1, 3, 4, 1, 6, -6, 2], [-1, 0, 5, 2, -1, 1, -3, 2, -1, 3, 2, -1, 0], [-5, -5, 2, -3, -1, -3, -3, 1, -2, 2, 4, -5, 2], [5, 2, 2, 2, 3, 5, 3, 1, -1, -1, -2, 3, 1], [-6, -2, 3, -3, -3, -1, 0, 4, 2, 2, 3, -3, 4], [-7, 2, -1, 1, 3, 2, -3, 3, 1, -1, -2, -1, -1], [2, 4, 3, 2, 2, 0, 1, -1, -2, -1, -2, 5, 2], [0, 3, -1, 1, 2, 4, -1, 2, 1, 4, 1, 0, 3], [-3, -6, 0, -3, 3, 0, 1, -2, -2, -2, -1, 2, 1], [1, -3, 1, 1, 1, 3, -1, 2, 0, 3, -3, 0, -1], [-2, 3, 0, -3, 0, 2, 2, -3, 0, 1, -1, 3, -3], [2, -6, 1, 1, -1, 3, -3, 3, 3, 2, -1, -2, -1], [1, 5, 1, 3, -1, 4, 4, 2, 0, -1, -3, 4, 1], [4, 2, 1, 3, -2, 0, 0, 1, 0, 0, 1, 4, 0], [0, 8, 1, 4, -2, 1, -2, 4, -2, -3, -4, 1, 0], [2, -4, -1, -2, -3, 0, 0, -1, -1, 3, 0, 0, 1], [-5, -1, 0, 4, 1, -2, -5, -1, 2, 2, 2, -3, -3], [-5, -2, -5, -1, 1, -1, 1, -2, 2, 0, 5, -3, -1], [-4, -1, 4, -1, -4, -1, 2, 0, -1, 1, 6, -1, -2], [-9, 5, -3, -8, -5, 1, 4, 2, 2, 2, -1, 0, 4], [3, 9, 0, 5, -3, -1, 1, -1, -1, -1, -1, 4, -3], [-5, 1, -2, -2, -1, -4, 1, -3, 0, 0, -4, -1, 1], [4, -7, 0, -3, 3, -5, -3, 1, 4, -1, 5, -5, 0], [-2, 6, 0, 0, 0, 1, 4, -3, -3, -3, -3, 6, 6], [5, 4, -1, 4, -4, 4, 6, 0, -4, -1, -4, 5, 0], [-11, -5, 1, -3, -5, -7, -3, 0, 4, 1, 5, -8, 1], [2, 4, 2, 9, 0, -3, -2, -5, -4, -2, 0, 2, -5], [2, -4, 1, -3, 5, 3, -3, -3, -2, 0, 2, 1, 1], [3, 0, 0, 1, 2, 2, 1, -2, -1, -2, -1, 2, 0], [1, 6, -3, -1, 3, 4, 5, -8, -1, -3, -2, 3, 0], [2, -4, 3, 5, 3, -3, -7, -3, -4, -3, -4, 1, -3], [-7, -2, 2, -5, 0, -1, 1, -1, 1, 1, 0, 4, 4], [-5, 4, -2, 0, -3, -5, 4, -3, -3, -4, 2, 4, 1], [0, -1, 6, -3, -3, 2, -1, -2, 2, 3, 4, -3, -1], [7, -4, 5, 5, -2, 0, -2, -2, -2, 2, 1, -3, -2], [1, -4, 0, -2, 3, 7, 2, 1, 1, 4, 3, 1, -1], [-2, -2, -2, 1, 0, 0, 0, 6, -1, 3, -7, -2, 4], [-2, -2, -5, 0, -2, 1, -1, -3, 1, -2, -1, -1, 0], [7, -4, 1, -2, -3, 7, -1, 5, 2, 4, 3, 0, 5], [-3, -4, 5, 1, 0, 2, -5, 0, -2, 1, -4, 1, -2], [-9, -5, 2, -5, 4, -5, -1, -2, 4, 2, 10, -9, 0], [4, -1, -1, 1, 0, 4, 2, -2, 2, 0, 0, -4, 0], [11, -1, 0, 3, -3, -2, 2, 0, -3, -1, -2, 1, 2], [2, 3, 1, 0, 3, 1, 5, -1, 0, -4, -1, 4, -1], [-4, -5, -6, -2, -3, -1, 0, 6, -1, 1, -1, -5, 3], [-1, 8, 3, 1, 2, 1, 1, 0, 1, -1, -1, 5, 0], [3, -6, 2, 1, 6, -4, 1, -10, 2, -1, 5, -2, -5], [-2, 3, -2, 7, 1, -2, 1, 3, -1, -2, -6, 3, 0], [-8, -2, 1, -1, 1, -2, 1, -3, -5, 1, 3, -1, 3], [9, -12, 3, 4, 1, 2, -4, 3, 5, 4, 4, -5, 1], [2, -6, 1, -2, 2, -1, -1, -5, -3, 0, 7, -1, 2], [-1, -1, 4, 3, 2, 6, 4, -1, -2, 2, -2, -1, 0], [-9, 1, 3, -13, 2, -1, 1, -4, 4, 2, 11, -4, 0], [-2, 8, 3, 2, -6, 1, 1, 1, -3, -4, 3, 7, 0], [-3, -9, -3, -4, 1, 2, -1, 0, 1, 1, 1, 0, -5], [3, 1, -2, 0, 5, 3, 3, -1, 1, 1, -3, 1, -1], [-1, -1, 0, -11, 0, 1, 4, 0, 7, 4, 3, -1, 2], [5, -7, 8, 2, -1, 4, -5, 2, 0, 4, 0, -3, 0], [5, 5, 5, 8, 0, 3, 1, -1, 0, 1, -6, 6, -2], [0, -4, -1, 2, 2, -3, -3, -1, 2, -2, -1, 1, 1], [0, -5, 1, -12, -6, 6, 1, 1, 4, 5, 5, -5, 4], [-5, -5, -2, 3, -6, -6, 0, 2, -2, -6, 0, 1, 0], [-1, 1, 1, 1, 3, 5, 1, 0, -5, 3, -3, 6, 5], [-9, -14, 2, 2, 3, -12, -8, 3, 0, -3, 5, -8, -4], [6, -4, 3, 4, 4, -4, -1, -5, 1, -1, -3, 0, -6], [-4, 3, -4, 8, -1, 2, 1, 4, 2, 0, -9, 5, 1], [2, -9, 3, 3, 2, 2, -2, 4, 4, 2, -6, -2, -7], [5, 6, -2, 2, 2, 7, 3, -3, -1, 0, -7, 2, 2], [-3, -2, -5, 1, 4, -3, 0, 1, 1, -2, -2, -1, -4], [-18, 5, 0, -1, -2, -4, 1, 2, 3, -2, 2, -1, -4], [7, -10, 1, 4, 0, 3, -3, -3, -5, 1, -8, 1, 0], [-3, -7, -4, -4, 3, -4, -4, 0, -1, -3, 3, -3, -1], [-7, -2, -4, -5, -2, -2, -1, -2, 3, 4, 4, -1, 0], [-6, -1, -2, 2, 5, -6, 2, -3, 0, -3, -8, 1, -6], [-9, -1, 4, -1, -9, -1, -1, 3, 1, 2, 0, 1, 3], [-4, -3, -5, 4, -4, 1, 3, -3, -3, -3, -4, 3, -2], [4, 2, -4, 1, 1, 6, 5, -7, 0, 3, -2, 2, -3], [-1, 4, 0, 3, -2, 3, -6, 9, -3, 0, -6, 1, 0], [10, 2, 0, -2, 4, 4, 2, -5, 0, 0, -4, 4, 3], [-6, 2, -9, 4, 6, -6, -3, 3, -2, -5, -1, 0, 0], [11, 1, 4, 6, 5, -2, 0, -6, -3, -5, 1, 5, -1], [-7, -11, 7, -3, -2, -2, -6, 5, 2, 2, 2, -7, 2], [6, -4, 0, 3, 6, 7, 0, 4, 0, 2, -2, 2, 3], [-3, 9, -6, 7, 2, -4, 3, 1, -1, -2, -7, -1, -2], [-1, 4, 3, 10, -1, -2, -6, -1, -8, -5, -7, 4, 3], [-11, -9, -1, -10, -4, -5, 0, -1, 6, 5, 13, -8, 4], [3, 8, 1, 1, 9, -4, 1, -4, -2, -8, -1, 6, -1], [-9, -9, 3, 0, 0, -2, -6, 7, -1, 1, 0, -4, -3], [-4, 6, -6, 0, -5, -4, 4, 1, -1, -1, -12, -2, 3], [-10, 6, 0, 7, 6, 1, -2, -6, -3, -4, -2, 6, -4], [0, 9, 0, 3, -5, -4, 5, 2, -2, -3, 4, 7, 1], [6, 0, 3, -1, 0, -1, 5, 0, 1, -1, -2, 5, 0], [-4, -5, 1, -7, -1, 4, 4, 1, 4, 2, 2, 2, 0], [-1, -5, 2, -5, 4, -3, -1, -1, 5, 5, 9, -12, 0], [4, 8, -7, 4, 0, 7, 3, -1, -1, 2, -5, 7, -1], [0, -7, 3, -2, -5, -5, -1, 9, 2, -1, 7, -8, 3], [-2, -10, 1, 2, -3, 5, 0, 6, -2, 4, 0, 1, -3], [9, -6, 0, -3, 0, -1, -6, 7, 0, 3, 4, -7, 2], [0, -4, -2, -2, -4, 7, -2, 1, 1, 3, 9, -2, 0], [-4, -3, -1, -7, -1, 4, 1, -5, 3, 4, 6, -1, 3], [-5, -11, 2, -3, -3, -2, 0, -1, 0, 1, 2, 3, -2], [13, 0, 3, -4, 3, 1, 1, -4, 4, -1, 3, 0, -1], [-9, 4, -1, 1, 5, -4, 1, -7, -5, -1, 2, -3, -3], [0, -6, 8, 3, -4, 0, -5, -1, -5, -2, 0, 8, 2], [-6, -6, -3, -11, -5, 2, 1, 6, 7, 4, 11, -7, 5], [-4, 3, -6, 2, 5, -4, -1, -3, -2, -7, -2, 2, -1], [-11, -3, -9, 3, -9, -7, -3, 2, -2, -3, 1, -4, -1], [-18, 0, 6, -8, 0, 1, -3, 3, -2, 6, 9, -3, 4], [-6, 2, -5, 6, -10, 4, 1, 9, 3, -2, -10, 5, 2], [12, 3, -2, 4, -7, 0, 2, 0, -5, -3, -12, 7, -3], [-19, 0, -2, 0, -1, -10, -5, 4, 2, 2, 4, -2, -4], [7, 0, -3, 2, 1, 2, 6, -5, -4, -3, -2, -1, 2], [13, -4, 6, 9, 0, 5, 3, 3, 1, -2, -2, 4, -6], [-1, -12, 10, -6, 2, 1, -6, -3, 4, 5, 14, -4, 2], [-3, -15, 7, 1, -6, -1, -8, 6, 4, 3, -2, -4, -4], [-7, -4, 7, 0, -2, -4, -4, 6, 2, 1, 10, -3, 6], [-4, -17, 6, -8, 4, -9, -8, -3, 5, -2, 7, -11, 1], [-7, -2, 9, 10, -4, -4, -3, -2, -3, -1, 7, -3, -6], [6, -8, 4, -1, -3, -3, -3, 4, 3, 8, 9, -2, 1], [-11, 0, 1, 0, -3, 6, 0, 11, -1, 5, -3, 2, 1], [-7, -4, 4, -9, 1, 0, -1, 1, -3, 0, 9, -4, -1], [-7, 0, -5, 8, -6, 2, 0, 3, -2, 2, -16, 3, -4], [16, 8, -5, -1, 5, 1, 4, -5, -3, -1, -2, 2, -1], [-6, 9, -11, 8, -5, -2, 2, 3, -1, -3, -3, -2, -1], [-10, -5, -1, -3, -9, -7, -1, 2, 3, 4, 8, -10, -1], [-8, 2, -2, -3, -1, 1, 1, -6, -6, 1, 2, 3, 5], [-2, -2, -3, 8, 0, 2, 1, 8, 4, 3, -4, -1, 2], [-6, -8, -2, 1, -3, 6, -3, 7, 2, 3, -5, 1, -1], [5, -10, 1, 4, 0, -4, -2, -2, -2, -1, 2, -3, 0], [-2, -6, 9, -8, -5, -2, -1, -3, 0, 7, 10, -7, 3], [2, -2, -8, -2, 10, 0, 6, -4, -1, -5, -3, 11, 0], [2, 4, -6, 10, 2, 0, -2, -7, -5, -5, -7, -5, 2], [-3, 1, 2, 3, 8, 2, 1, -4, -9, -5, -5, 5, -6], [6, 6, -2, 0, 1, 4, 5, 2, -8, 1, 1, 8, -1], [5, -13, 4, -3, 1, 6, -4, -3, -2, 5, 6, -3, 1], [-2, -6, 6, -7, 6, 2, 2, -4, 6, -1, 7, -4, -2], [-1, 3, -5, 7, -2, 4, -1, -5, -7, -1, -6, 8, 0], [-15, 12, -4, 0, -4, 1, 2, 5, -5, -5, -2, 2, 4], [8, 1, -11, 4, 1, -2, 2, 0, 1, -7, -13, 2, -3], [2, 8, -3, 2, -3, 1, 3, 1, 7, -1, 0, 0, -3], [7, 16, -10, 3, 1, 3, 10, -6, -6, -1, -3, 10, -2], [-3, 2, -1, 3, -1, -8, 1, -8, -2, 0, -7, 2, 1], [-5, -11, 4, -7, -5, -9, 5, 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1, 6, -9, 5, -4, 4, -5, 8, 7, 4, -3, -7], [-3, 13, -3, 6, -5, 13, 6, -3, -8, 4, 1, 0, 0], [-12, 18, -9, 23, 10, -4, 7, -6, -14, -13, -11, 15, -4], [12, 2, 3, 2, 0, 6, -1, 1, 4, -1, 6, 7, 2], [-27, -8, 5, 1, 1, -2, -5, 5, 2, 0, 12, -5, 0], [15, 19, -4, 12, 2, -3, 13, -1, -5, -8, -11, 13, -8], [0, 2, -11, 5, 4, 1, -8, -6, 1, 1, 1, -11, -4], [26, 9, -12, 8, 11, 8, 3, 4, 4, 4, -11, -1, -1], [13, -23, 5, -11, -4, -2, -4, 2, 10, 11, 11, -12, 7], [-11, 9, 9, -1, -8, 5, -2, 10, -3, 0, -3, 1, 4], [-8, -10, 4, -3, 0, -8, -13, -3, 11, 6, 1, -11, -6], [-1, 6, 6, 5, -4, -3, 0, -1, 2, -2, 3, 6, 6], [10, 14, 1, -5, 5, 0, -1, -5, 8, 5, 13, -3, -9], [-13, -3, -11, -1, -4, -5, 0, -12, 0, -6, -4, -4, -2], [-26, 5, -7, -7, -18, -11, -1, 12, 0, -1, 4, -10, 15], [13, -8, 8, 4, 9, -7, -2, -11, 5, -6, 1, -1, -11], [2, 15, -3, 7, -10, 9, 5, 5, -2, -2, -4, 0, 3], [14, -6, 7, 3, -12, 4, -3, 11, -6, 0, -7, 1, 4], [4, 16, -16, 14, 12, 10, 14, -3, -1, -4, -10, 8, -4], [16, 8, -6, 16, -14, 5, 9, -3, -4, 0, -10, 2, -6], [9, -4, 7, -4, 6, 6, -8, 9, 1, 2, 2, 9, 0], [-2, -7, 7, 3, 2, 1, -5, 5, 2, 8, -8, 0, -7], [-2, -7, 2, -8, 10, 4, 3, 6, 11, -2, 4, 4, -2], [26, 5, 5, 4, 1, -1, 11, 2, 1, -7, 7, 4, -4], [13, 5, -6, -2, -7, 8, -2, -3, -5, -1, -1, -7, 3], [-4, -25, -1, -10, 8, -5, -3, -6, 7, -4, 13, -6, -1], [-3, -22, 15, -16, -12, 4, -6, -1, 4, 16, 28, -17, 9], [29, -8, -2, 13, 14, 7, 4, -7, 2, 0, -2, 5, -8], [21, 7, -4, 11, -1, 2, 4, -9, 7, 3, -12, 1, -1], [0, -13, 9, -5, 4, -5, -4, -15, 2, 0, 10, -7, 2], [-15, -12, -6, -17, 4, -8, -4, 4, 18, 16, 10, -16, 2], [-19, 0, -2, -16, -3, -3, 1, 0, 1, 1, -3, -6, 9], [25, 1, 15, -11, 4, 3, 1, -18, 6, 3, 4, 1, 2], [-12, 1, -10, 0, -1, -5, 0, 3, -3, -6, -21, 3, 1], [-7, 1, -7, 16, 0, 2, 4, -5, 3, -5, -16, -4, -8], [-23, 17, 8, 0, -5, -7, -6, -2, -6, 0, 5, 2, 9], [12, -3, 6, -23, -2, 1, 7, -2, 12, 5, 5, 1, 2], [15, 26, -7, 2, 2, 16, 10, -6, -4, -3, -14, 15, 8], [6, 4, -5, -10, -3, 10, -4, 19, 9, 10, 1, -7, 11], [13, -9, -4, 4, 7, 5, 6, -7, -2, 1, -2, 7, 7], [-3, 5, 2, -12, 10, -5, -4, 3, 3, -2, 0, 5, 10], [2, -4, -9, -9, 0, 5, 5, -4, 7, 1, -4, 5, 2], [1, 2, -7, -2, -7, -1, 0, 9, 1, -6, -5, 13, 4], [7, 6, -5, -2, 0, 7, -1, -3, -2, -1, -8, 17, 12], [12, 9, -15, 11, 6, -5, 10, -7, -7, -16, -3, 14, -6], [4, 16, -9, 20, -19, 1, 7, -4, -10, -4, -15, 9, -5], [8, -6, -9, -8, 1, -3, 3, 10, 4, -2, -7, 3, 2], [-5, -4, -2, 9, 2, 7, 4, 6, 0, -7, -19, 14, -8], [9, 13, 2, 10, -2, 2, 1, 6, -8, 3, -16, 7, 0], [2, -4, -7, -7, 7, -6, 13, -7, 5, -5, 19, -7, -7], [6, 3, 2, 4, 0, 7, -3, 9, -9, 4, -20, 9, 14], [18, 14, 0, 0, 5, 0, 4, 2, 2, 1, 2, 8, -9], [9, -11, 9, 6, 1, -6, -1, 5, 9, 4, 9, -12, -6], [-21, 27, -17, 6, -5, -2, -1, 9, -12, -7, -17, 0, 9], [16, 17, -3, 2, -2, 0, 14, 1, -2, -9, -5, 14, -2], [-15, -8, 1, -17, -5, 5, -4, 14, 5, 3, 1, -8, 11], [-2, -6, 0, 6, 5, -6, -10, 9, 11, 4, 14, -14, -8], [-8, -4, 8, -10, -2, -6, -12, -8, 6, 6, 6, -6, 4], [-2, -19, 3, -4, -4, 5, 0, 3, -13, 3, 4, -6, 8], [16, -1, 7, -6, -4, 1, -3, 1, 9, 17, 9, -8, 4], [-8, -4, -5, -5, -2, -7, -3, 2, 12, -1, 4, -11, -2], [24, -4, 14, -5, 13, 3, -4, 0, 2, 3, 20, -6, -4], [11, 4, 4, 2, 8, -2, -7, -11, -8, -2, 13, 7, 6], [-17, 8, -2, -5, -5, -1, -2, 5, 8, 7, -13, -6, 1], [2, 8, 6, 22, -19, -6, -5, 7, 3, 6, -1, -8, -7], [14, -19, 5, 0, -5, 0, 0, 7, 6, 1, -3, -8, -5], [-14, 18, -5, -3, -15, 3, 5, 7, -1, 1, -6, 15, 7], [-35, -5, -5, 2, 8, -2, -1, -1, 1, -1, 9, -4, 0], [-26, 7, -1, -2, 1, 2, 5, -2, -12, -7, -2, 2, 12], [7, -3, 3, 16, -2, -4, -1, 6, -4, -6, -15, 10, 2], [-7, 35, -4, 7, -5, 7, 4, 12, -10, -3, 1, 3, 2], [7, 16, 14, -6, -2, 17, 3, 5, -1, 5, 7, 10, 8], [-19, 15, 0, -3, 7, -1, -5, -7, 2, 0, -8, -5, -2], [5, -4, -5, -20, 5, 1, 8, -14, 5, 6, 5, 3, -3], [27, -7, 10, 20, 2, 5, -5, 10, -3, 1, -15, 3, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8001_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8001_2_a_o();".
// 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_8001_2_a_o(:prec:=13)
    chi := MakeCharacter_8001_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_8001_2_a_o();".
// 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_8001_2_a_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8001_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![8, -12, -116, 146, 372, -211, -467, 61, 242, 19, -53, -10, 4, 1]>,<5,R![16, -184, 412, 784, -1990, -1590, 1877, 1492, -340, -457, -50, 39, 12, 1]>],Snew);
    return Vf;
end function;