
// Make newform 8001.2.a.n in Magma, downloaded from the LMFDB on 20 April 2024.
// 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_n();".
// 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_n();".
// 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 := [-1, 7, 16, -71, -39, 159, 44, -123, -11, 41, -3, -5, 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], [24, -664, 95, 1974, 474, -1552, -315, 551, 21, -78, 9, 1], [13, -114, 1210, -388, -3981, -193, 2903, 50, -868, 75, 97, -19], [-69, -302, 1628, 1544, -4093, -2037, 3393, 652, -1191, 40, 150, -28], [117, 180, 840, 997, -3669, -2340, 3615, 986, -1447, 5, 203, -37], [-207, 568, 2271, -3006, -5043, 2867, 3077, -1260, -692, 254, 48, -17], [73, -635, -2204, 3006, 5043, -2867, -3077, 1260, 692, -254, -48, 17], [-60, 856, 1337, -3193, -5071, 2205, 4104, -875, -1292, 262, 145, -36], [223, 39, -2967, 905, 7034, -641, -5163, 466, 1577, -239, -176, 40], [-61, 102, 2553, -679, -7419, 282, 5968, -415, -1921, 282, 220, -50], [-76, -19, 1966, -1025, -7062, -21, 6190, -81, -2177, 247, 273, -59]];
    Rf_basisdens := [1, 1, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67];
    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)
    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+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 := [[1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 1, 0, 1, 1, 1, 1, 1, -1, -1, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, -1, 1, 0, -1, -1, -1, 0, 1, 0, 0], [-1, 0, 0, 0, 1, 1, 1, 1, 1, -1, 0, -1], [1, 0, 0, 0, 1, -1, 0, -1, 0, 0, -1, -1], [-1, 2, 2, 0, 2, 1, 0, 0, 1, 0, -1, 1], [2, 0, -1, -1, -2, 2, 1, 0, 0, 0, 0, 1], [2, 0, 1, -1, 2, 0, 2, 1, 2, -1, -2, -1], [-1, -2, 1, 0, -1, 0, 3, 0, -2, 0, 1, 0], [1, -2, 0, 3, 1, -2, 0, 1, 1, 0, 2, -2], [0, -2, 1, -2, 0, 1, 2, 1, -4, -1, 0, 1], [2, 0, 0, 0, 1, 0, -1, 0, 3, 0, -2, -1], [2, 0, -1, 1, 0, 0, -1, 1, -1, 1, 2, 1], [3, 1, 1, 1, 2, -1, -2, 1, 1, -1, 0, 0], [1, 0, 0, 2, 0, -1, -2, 0, -2, 1, 0, -1], [0, 2, 2, 1, 3, -1, 1, 0, 0, -2, -2, 0], [1, 0, 0, -1, 2, -1, 3, 1, 3, -3, -2, -1], [7, 0, 2, 3, 1, -2, -2, 0, 5, -3, 1, -2], [-2, 1, -1, -1, -1, 2, 0, 0, 0, -3, 0, 1], [2, -1, 0, 2, -3, -1, 0, -2, -2, 3, -2, 1], [2, -3, -2, -4, 1, 1, 2, -2, -2, 0, -2, 1], [0, 1, 1, 2, -3, 1, 2, 1, -5, 6, 1, 4], [-1, 1, 2, -1, 0, -2, 2, 0, -6, 2, 3, 3], [-1, -2, 2, -1, -1, 2, 3, 2, 5, -1, 1, -4], [0, 1, -3, 1, 1, -1, -4, 0, -2, 3, 1, 0], [-1, -2, 0, 1, -2, -3, 0, 1, -4, 2, 4, -3], [-3, 3, 1, 0, 1, -1, -1, -1, 0, 3, 2, -1], [8, 3, -4, 0, 2, -2, -1, -3, 2, 3, -6, 3], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 3, 1, 2, 4, 0, 0, 2, -2, 2, 0, 2], [6, -2, 4, 1, 4, 0, 0, 3, 5, -3, 2, -2], [-2, 5, -1, -1, 6, -3, -4, 1, 5, 1, -4, -4], [5, 2, 5, 2, -1, 3, 1, 0, 1, 2, -1, 4], [-3, -1, -3, -5, -2, -1, -1, -5, -7, 2, -3, 4], [4, -2, 2, -1, -5, 3, 3, -1, 0, 3, 2, 1], [2, 1, 1, 2, -2, -2, -4, -7, 0, 2, 0, 2], [7, 0, -3, 4, -2, -3, -4, -9, -1, 3, 2, 1], [-4, 2, 1, -2, 3, 0, 3, 0, 6, -6, -3, -4], [3, -2, 3, 2, 4, -3, 0, 0, 7, -4, 1, -7], [8, -6, -6, 0, -7, -4, -3, -4, -5, 4, 4, 1], [6, 3, 1, 1, 0, -4, -4, -5, -1, 0, 3, 0], [-1, 3, 3, 1, 1, 1, -4, -2, 3, -5, 3, 1], [1, -1, -8, 3, -2, -6, -6, -2, -5, 5, 2, -3], [-7, 0, 0, 0, 1, 2, 0, 4, -2, 0, -2, -2], [5, -4, -2, -1, -8, 1, -1, -3, -8, 1, -1, 3], [4, 2, -1, 1, 5, -3, -1, -1, 2, 4, -5, -1], [4, 2, -1, 3, 3, 1, -2, 1, 4, 5, 0, -2], [0, -1, -1, -2, 0, -1, -1, -8, -6, 0, 3, 1], [2, 6, 1, -1, 5, 1, 3, 3, 0, 0, -2, 2], [6, -3, -3, 1, -6, -3, -3, 1, -9, 8, 9, 5], [-5, 0, 2, 0, -1, 1, -3, 0, 0, 3, 2, -3], [-5, 4, 3, -3, 0, 2, -3, -3, 3, -6, -1, -1], [4, 0, -3, 2, -3, -5, -7, -5, -2, 6, 6, 3], [-3, 2, -3, -4, 1, 4, 3, 4, -2, -6, -4, -1], [-4, -3, 0, -2, -4, -2, -2, -6, -8, 4, 3, 2], [-12, 6, 4, 0, 1, 3, 1, 0, -6, 7, 3, 3], [7, -4, -2, 1, -5, 2, -3, -1, 0, 1, -1, -1], [7, -1, 0, 4, 4, -6, 2, -1, 2, -3, 2, -3], [0, -1, 2, 4, 10, -1, -3, 2, 6, -5, -1, -3], [-2, -3, 6, 3, -2, -1, 0, -2, 2, 1, 4, -2], [-2, -5, -1, -2, 1, -4, -4, 2, -5, -3, 3, -4], [4, -1, 2, 0, 2, 2, 3, -3, -4, 2, 1, 1], [3, 1, -1, 2, -7, 5, -1, 2, -5, 2, 5, 6], [7, 7, 1, 3, 5, -6, -6, 0, 7, -5, -1, 0], [-6, -8, 3, 1, 3, -3, 4, 7, 7, -6, 3, -8], [5, 2, -8, 0, -2, -9, -3, -4, -4, 2, 2, 4], [4, 1, 4, 3, -2, 0, 3, -4, -1, 1, 1, -3], [7, 4, 5, 10, -2, -2, -8, -4, 7, 2, 1, -4], [-10, 7, 1, -2, -2, -1, -1, -3, -4, 3, 1, 0], [12, -10, -5, -9, -10, 2, 1, -5, -4, 2, -3, 0], [3, -11, -5, -5, -4, 4, -2, -8, 2, 2, -5, 3], [-7, -1, 1, -5, 6, 3, 4, 6, -1, 1, -1, -1], [-9, 10, 7, 4, 11, -2, -1, 7, 10, -7, 0, -8], [-5, 1, 3, 6, 3, -4, -3, 6, 3, -4, 1, -8], [-1, 1, 1, 0, 3, 4, -5, -1, 7, -1, -8, -2], [-2, -2, -7, -10, -4, 2, 4, 0, -12, 5, 0, 4], [-7, 1, 3, -6, 3, 2, 2, -2, -13, 3, 3, 6], [0, -4, 5, 5, -5, -3, 0, 2, -6, 6, 7, -3], [4, -1, -6, -5, 0, -6, -9, -7, -4, 6, 2, -1], [4, 6, -9, 2, -2, -2, -5, 2, -3, 5, -1, 6], [10, 3, 8, 1, -1, 5, 4, 2, 2, -2, -5, -2], [8, -7, -6, -2, -11, -3, -4, -8, -2, -1, 2, -5], [-3, -2, -3, 1, -11, 1, 0, 4, -3, 0, 5, 1], [0, 1, 0, 0, 6, 4, 4, 11, 6, -2, 0, 1], [5, 1, 4, 3, 2, -3, -3, -1, 1, -5, -5, -4], [0, -3, 2, -1, -7, 3, 5, 0, 6, -6, 3, -2], [-5, 10, -1, 3, 0, -4, -7, -1, 2, 1, 0, 0], [10, -3, -3, -1, -1, -3, -3, 0, 6, 6, -1, -4], [-2, 8, 5, 6, 6, 2, -2, 5, 7, -5, 0, -2], [11, -3, -1, -3, -6, 6, 6, -2, 0, 2, -5, 4], [4, -5, 4, -6, -2, 5, 4, 3, 11, -5, -1, -2], [-1, -2, -4, -9, -8, 7, 8, -6, -6, -5, -3, 5], [-4, 2, -3, 5, -9, 0, -4, 0, -4, 13, 1, 7], [-3, 3, 2, 1, -8, 6, 2, 4, 8, -2, 6, -4], [9, -7, -7, -6, 1, -3, -7, -13, -2, 6, -8, 5], [4, -4, -5, 1, -13, 0, 1, -4, -7, 7, 9, 5], [10, 12, -4, -4, -2, 2, -5, -8, 5, -4, -13, 7], [-4, 1, 9, 0, -1, -3, -1, 1, -3, -8, 4, -2], [-6, 9, 6, -4, 5, -1, 5, 4, -4, -2, 4, 5], [-7, -7, -5, -2, -7, 1, -7, -5, -4, 0, 6, 3], [6, 1, -2, 9, 1, -12, -6, 1, 5, 6, 2, -6], [3, -3, 2, -7, -5, 5, 7, -2, -15, -2, 7, 9], [0, 5, 2, 7, 2, -1, -6, -2, 4, -2, -5, -1], [1, 2, 1, 4, -2, 11, 5, 7, -2, 6, -3, -1], [-13, 13, 11, 7, 8, -1, -6, -3, 12, -5, -1, -6], [-15, -2, 0, -7, -11, 7, -3, 3, -7, 1, 3, 6], [5, 5, 1, 5, 0, 3, -6, -2, -2, 5, -3, 2], [0, -10, -2, 0, 8, -11, -2, -3, 2, -3, -6, -10], [-16, 4, 4, 0, -2, 8, 6, -2, -3, 2, 6, 7], [3, 5, -2, 4, 2, 0, -8, -7, -6, 4, 7, 1], [16, -2, -1, -2, -5, 3, -1, -1, 5, -4, -9, 2], [4, -7, -3, -10, -3, 6, -1, -7, 7, -2, -7, 0], [-8, -2, -9, -5, -9, 1, 1, -1, -10, -2, -3, 1], [5, 10, 2, 2, 7, 4, 6, 3, 10, 0, -10, 5], [-1, 4, 4, -1, -1, 3, 7, 9, 0, 0, 4, -7], [-9, 7, 6, -2, 0, 10, 12, 10, -2, -5, 1, 1], [-6, -1, 8, 3, -5, 7, 1, 2, 4, 4, 10, 0], [3, 0, 6, -2, 10, -1, 8, 10, 3, 3, -3, -1], [3, 2, 2, -1, 3, -2, 4, 2, -2, -2, -10, 2], [2, -5, -1, 1, 3, 4, 10, 11, 5, -1, -1, -4], [3, -6, -8, 0, -6, -2, -3, -3, -4, 1, 0, -3], [-7, 2, -1, -7, -7, 18, 12, 8, 0, -12, -1, 10], [-5, 2, 1, 4, 2, 0, 8, 12, 10, -1, 2, -11], [-3, 7, 8, 6, 4, 0, 6, 10, 8, -3, -2, -7], [-2, 6, -5, -8, -2, 2, 5, -1, -14, 13, -10, 11], [5, 3, -3, -5, 1, 8, 7, 3, 4, -4, -14, 3], [0, 7, -4, 0, 2, -7, -2, 0, -3, 9, -3, 2], [-2, 11, 6, 2, 15, 2, 3, 11, 20, -11, -4, -6], [1, 3, 7, 4, 2, 4, 4, 3, 9, 1, 1, -7], [4, -1, -4, 2, -1, -1, -2, -8, -17, 10, -4, 6], [-10, 9, 2, -7, 4, 7, 9, 8, 5, 1, 2, 7], [-11, 8, 4, -2, 11, -1, -4, 1, 1, -9, -3, -3], [-5, -3, 0, 11, -1, 2, -3, 5, 7, -3, 12, -7], [9, 9, -2, 3, -4, -4, -6, -13, -10, 2, 0, 3], [-8, 6, -12, 2, 0, -5, -15, -7, -1, 0, -11, 4], [0, -4, 2, -1, -2, 0, 2, 3, -6, -2, 4, -1], [0, -15, 5, -6, -3, 3, 9, 0, 3, -5, 1, -5], [-11, -3, -2, -8, -1, -3, 1, 6, -2, -3, 0, -7], [14, 13, -1, 3, 9, 0, 0, 3, 9, -3, 1, 3], [-5, 6, -1, 6, -2, -2, 1, 8, -2, 11, 2, 5], [-3, 9, 3, -2, 8, -6, -5, 1, 6, -8, 3, -9], [8, -12, -3, -3, -1, -3, 4, -5, 1, -1, 1, -5], [24, 1, -4, 8, 2, -5, -1, -3, 11, 2, -6, 0], [3, -8, -7, 0, -16, -1, -3, -3, -16, 11, 14, -2], [7, -1, 2, 2, 0, 2, -2, -2, 1, -1, -1, 3], [-10, 2, -8, -4, -10, 1, -9, 2, -2, 9, 0, 7], [7, -1, 11, 1, 8, 8, 13, 7, 14, -15, -8, -4], [-15, 7, 7, 2, 3, 10, 10, 6, 11, -10, 0, -7], [17, -7, -2, -6, -12, 2, 4, 4, -5, 5, 3, 12], [0, 8, -10, -5, -9, -2, -3, -6, -12, 12, -4, 13], [-2, 5, -5, -6, -1, -3, 3, -1, -2, 3, 6, -3], [11, -12, -1, -8, -6, -2, -8, -12, -10, 3, -1, 3], [4, 9, 8, 6, 4, 3, 8, 3, 0, -1, -8, 3], [-2, 9, -4, -9, 9, -2, 2, 4, 3, -4, -9, 2], [-10, 0, 3, -9, 5, 12, 10, 4, -7, -9, -11, 4], [16, 1, -3, 0, 2, 1, 3, -1, -3, 10, -3, 5], [2, -2, 1, 2, -6, -7, -5, -7, -9, 0, 7, -3], [-1, 1, -4, 7, 9, -9, -7, -4, -6, 4, -1, 3], [8, 5, 3, 13, -6, -2, -6, 6, 6, 2, 10, -2], [11, -17, -9, -7, -14, 2, 1, -9, -14, 4, 7, 11], [-7, 0, -5, 2, -14, -2, -17, -10, -6, 10, 18, 3], [-19, 5, 2, -7, -4, 10, 13, 3, -1, -7, 4, 8], [0, -5, -10, -1, -4, -7, -12, -12, 2, 5, 5, -8], [-16, 12, 10, -7, 5, 7, 9, 5, 6, -7, -8, -3], [-2, -4, 4, -6, 11, 9, 11, 2, -10, 1, -2, 5], [-19, 2, 5, -5, 7, 4, 3, 9, 11, -7, -2, -12], [-3, 7, 14, 5, 11, 1, 7, 13, 5, -2, 0, -7], [6, 1, -6, -1, -3, -4, -5, 0, 6, 4, 1, -10], [1, -5, 1, 0, -7, 0, 6, -2, -7, -1, 22, 5], [4, 1, 2, 12, 9, -11, 1, 1, 8, 5, 8, -5], [3, -8, 7, -5, -1, 5, 7, -3, -14, 1, 0, 10], [-6, 8, -5, 2, -4, 2, -7, 5, -8, 0, 0, 9], [-6, 8, -7, 1, 2, -4, -14, -4, 2, 2, -12, 1], [-13, 10, 15, 9, 7, -2, 1, 8, 4, -7, 10, -10], [-17, 6, 4, 0, 8, 13, 13, 12, 6, -6, -3, -5], [-1, 9, 10, 9, 13, -1, -3, 6, 9, -2, 0, -2], [7, 4, -5, -5, 6, -2, -4, -1, -2, 1, -13, -1], [-1, -10, -4, 3, -11, -8, -7, -3, -2, 13, 3, 1], [0, 0, -3, 3, -12, -11, -1, -8, -16, 10, 8, 9], [7, 8, -5, 4, -13, 2, -4, -7, -15, 6, 6, 14], [-4, -9, -1, 1, -2, -13, -2, -10, -5, 0, 4, 0], [-7, -3, 0, -1, 9, -6, 1, -3, -5, 4, 7, -3], [18, -17, -10, -13, -10, -2, -6, -10, -5, 3, 1, 5], [1, 12, 3, -1, 4, 1, -9, -3, 14, -4, -3, 6], [10, -11, -1, 1, -2, 2, -12, -1, 1, -4, 1, -6], [-7, -5, 4, 10, 3, -4, -8, 9, -1, -3, 11, -7], [-10, 3, 0, 7, -4, -3, -16, -6, -10, 1, 2, 9], [13, -6, -9, 7, -3, -3, 2, 3, -2, 15, 8, 5], [2, -12, 3, 0, 7, -13, 12, 8, 9, -3, -5, -19], [-9, 0, -3, 9, -11, -3, -16, -2, 4, -2, 1, -10], [6, 8, -4, 8, 0, 2, 2, 12, 4, 3, 8, 2], [1, 16, -1, 6, 1, -8, -10, 1, -8, 8, 14, -4], [-3, -15, 1, -4, -2, 4, 0, 0, -7, 1, 6, 4], [-2, -15, -4, -11, -6, 2, 8, 3, -3, 2, -1, -3], [0, 0, -3, 7, -1, -11, -5, -11, 4, 10, -2, 0], [-5, 14, -1, -1, 4, 6, 2, -4, 8, 3, -17, 4], [-3, 7, -4, 14, 11, -17, -20, -3, 8, 8, 3, -14], [-8, 6, 10, 5, 0, -1, 0, 14, 13, -13, -4, -13], [-6, -8, 0, -6, -3, 1, 2, -1, 1, -5, 10, 3], [15, -13, 13, 10, 4, -10, 1, 0, 17, -8, 12, -7], [-16, 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-5, 8, 4, 5, -19, -8, -6], [4, 23, 7, -11, -2, 17, 13, 5, 16, -20, -11, -6], [-10, -9, 7, -9, -21, 27, 19, 7, -9, -4, 14, 7], [30, 5, 4, 3, 9, -5, -19, -14, 8, 12, -7, 8], [-28, 3, 3, 10, -11, 8, -15, 2, -12, 1, 13, 0], [17, -8, 14, 3, 18, -7, 17, 26, 21, -12, -7, -2], [-12, -11, -7, -10, -6, 0, 11, 23, -7, -4, 6, 4], [7, -38, 3, 0, -3, 0, 0, -1, -1, -10, 15, -2], [10, -1, -6, 4, -7, 2, 1, 2, -11, 8, 13, 7], [12, -1, -3, -34, 15, 9, 21, -1, 7, -12, -15, 5], [-35, -9, -3, 9, -3, -2, 10, 9, -15, 7, 2, -10], [-20, -3, 22, 15, 15, 11, 21, 26, 16, -10, 21, -1], [-13, -2, 2, 5, 13, -3, 2, 5, 26, -18, -8, -22], [32, -16, 9, -7, -2, 6, 10, 2, -5, -5, -10, -5], [1, 7, 4, -6, 2, 9, 14, 9, -14, 1, -5, 16], [17, 11, 0, -17, 11, 0, 4, 0, -3, -10, -14, 2], [31, -14, 11, 0, 10, -9, -10, -12, 5, 15, 3, -6], [1, -7, 21, -9, 15, 14, 29, 29, 25, -15, -14, -4], [-4, -11, -25, -18, -3, -2, -19, -1, -14, 10, -7, 1], [-17, -9, -19, -17, -11, 3, 16, -1, -25, 3, -9, 20], [8, -8, 0, 2, -2, -5, -16, -5, -6, 7, 1, 10], [-12, -8, 3, 7, 13, 2, -2, 11, 35, -32, -6, -19], [-25, 2, 6, -2, 3, 13, 10, 18, 23, -21, -10, -18], [-34, -4, -6, -16, 7, 12, 10, 17, -11, -10, -5, 3], [-17, 20, 11, -13, 14, 18, -1, 2, 6, -19, -1, 20], [15, -1, -7, 4, -9, -7, 7, 9, 8, -13, -12, -5], [30, -22, 4, -5, -9, 4, 17, 5, 3, 7, -7, 14], [-8, -18, -15, -15, -17, 12, -2, -5, 1, 1, -8, 15], [-17, -5, -14, -20, -23, 7, 9, -15, -25, 7, -12, 25], [-25, 3, 14, -6, -5, 6, 27, 10, -11, 1, 14, 7], [15, 3, -8, 3, -15, 8, 1, -3, 1, -5, 12, -10], [7, 14, -4, -14, -13, 6, -1, -16, -29, 11, 2, 18], [10, -11, -1, -9, -4, 5, -12, -18, -4, -16, -8, -2], [8, 5, 16, 7, 12, 20, 14, 18, 14, -23, -1, 2], [-7, 12, 6, -5, 22, 4, 2, 7, 13, 1, -17, -6], [-13, 3, 15, 10, 10, 6, 3, 27, 13, 8, 0, 3], [-21, 15, 19, 4, 18, 19, 22, 20, 23, -10, -13, -3], [-5, -5, -24, 6, -15, -8, -8, -1, -12, 24, 28, 3], [-14, 0, 2, -1, 14, -7, 0, -11, -31, 7, -8, 7], [-4, -20, 3, 17, -14, -8, -16, -6, -19, 28, 29, -6], [3, 5, 15, -1, 11, -7, -6, -20, 3, -15, -2, -14], [0, 21, 11, 12, 35, -12, -2, 17, 36, -14, -19, -10], [-24, 2, 1, 5, -9, 1, -3, -2, -24, 11, 23, 2], [-18, -8, 5, -23, -1, 26, 30, 14, -3, -25, -5, -1], [-9, 0, 19, -5, 7, 18, 16, 11, 10, -26, -17, -2], [-1, 26, -8, 5, 6, 4, -6, -2, 8, 16, -1, 11], [-19, 8, 19, 3, -1, 13, 30, 19, 12, -4, -10, -1], [25, -19, -18, 6, -8, -13, -4, -7, -20, 28, 2, 16], [21, -5, 3, -9, -2, 6, 12, -14, -19, -1, -1, 9], [24, -2, -1, 21, 3, -18, -20, -9, 31, -9, 16, -12], [12, -15, -11, 0, -5, -4, -14, 1, -7, -1, -10, 3], [-26, -10, 4, 3, 4, -1, -3, 10, -7, 0, 9, -2], [-13, -14, 18, 4, -10, -9, 16, 6, -19, -1, 16, 2], [-18, 15, 14, 16, -3, -8, -3, -1, -15, 5, 16, 2], [-4, -5, -4, 13, 6, 1, -4, 17, 21, -15, -1, -14], [-5, 29, 2, 5, 11, 4, -5, 18, 25, 0, -28, -3], [-8, 0, 7, -11, 5, 11, 8, 25, 14, -8, 9, 3], [-42, -10, 13, -3, -2, 9, 6, -5, 8, -13, -2, -2], [-10, -4, -14, 4, 13, -11, -1, 5, -15, 3, -10, -8], [20, 9, -4, 6, 5, -7, -12, -7, 21, 16, -14, -11], [6, -2, -7, 13, 1, -23, -25, -5, 2, 21, 6, -8], [25, -7, -24, -2, -8, -17, -7, -6, -10, 15, -9, -11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8001_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8001_2_a_n();".
// 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_n(:prec:=12)
    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_n();".
// 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_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8001_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![15, -76, 60, 199, -278, -116, 303, -47, -107, 44, 8, -7, 1]>,<5,R![111, -1165, 1390, 1888, -3700, 633, 1639, -753, -164, 139, -7, -7, 1]>],Snew);
    return Vf;
end function;