
// Make newform 8001.2.a.p in Magma, downloaded from the LMFDB on 25 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_p();".
// 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_p();".
// 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 := [44, -42, -418, 301, 1134, -875, -1067, 959, 331, -414, -12, 76, -9, -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, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1454, 274, -5045, -486, 7866, 495, -7387, -58, 3055, -60, -541, 21, 34, -2], [-2510, 1918, 11241, -7437, -7878, 6669, -4044, -1581, 3945, -223, -917, 126, 67, -12], [-2210, 0, 17433, -2812, -32354, 8065, 23317, -6544, -7671, 2244, 1163, -347, -66, 20], [-135, -2055, 2173, 13845, -10851, -18767, 14382, 7501, -6637, -843, 1256, -68, -84, 13], [896, 5480, -9882, -17651, 31707, 12130, -34361, 2852, 14731, -3791, -2711, 909, 180, -67], [2190, -9864, -12311, 44380, 13591, -54117, -819, 25379, -2268, -5322, 662, 479, -53, -13], [1254, 4658, -13283, -17863, 39299, 12625, -41748, 2794, 17786, -3851, -3252, 930, 214, -69], [-2230, 1370, 12965, -11386, -21114, 18183, 14853, -11410, -5006, 3413, 796, -489, -48, 27], [-2091, 411, 14453, 2185, -31828, -584, 29235, -3909, -11643, 2570, 2052, -557, -132, 40], [4160, -16166, -17463, 77194, -5769, -92973, 37737, 36551, -21938, -4498, 4548, -193, -319, 51], [-2026, 4110, 18836, -23037, -51096, 40444, 52359, -32003, -21914, 11683, 3964, -1945, -259, 120]];
    Rf_basisdens := [1, 1, 1, 274, 274, 274, 137, 274, 274, 274, 137, 137, 274, 274];
    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 := [[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, 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, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [-1, 0, 0, 1, -1, 0, 1, 0, 0, -1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, -1], [0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0], [1, 0, -1, -1, 0, -1, 0, -2, 0, 1, -1, 1, -1, 1], [2, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, -1, -1, 0], [-2, 0, 1, 3, -1, 1, 2, 2, 0, -1, 1, -1, 0, -1], [0, -1, 3, 1, -2, 2, 1, 1, -1, 0, 0, -1, 1, -1], [-2, 2, -1, 1, -2, 1, 2, -1, -2, -1, 0, 1, 1, 1], [1, -1, 1, 0, -1, 0, 0, -1, -2, -1, 0, 0, 2, 0], [4, -2, 1, 0, -1, 0, -1, 1, -1, -2, 1, -1, 0, -2], [0, 1, -3, -2, -1, -2, 1, -1, -1, 0, -1, 1, -1, 0], [2, 1, -1, -2, -1, 0, 0, -3, -2, 0, 0, 1, 0, -2], [-4, 1, 1, 0, 1, -1, -1, 2, 2, 0, 0, -1, 0, -1], [3, -1, 0, 2, 0, 2, 2, 2, -1, -1, 2, 0, 0, -1], [1, 0, 2, 3, 0, 2, 1, 3, -1, -1, 1, 0, 0, 0], [0, 1, -1, -1, 0, 0, -2, -3, 0, 2, -3, 1, 0, 1], [2, -1, 3, 4, -2, 2, 2, 2, 0, -2, 2, -2, -1, 0], [2, -1, 0, 0, 0, 0, 1, 2, -3, -2, 1, 0, 2, -1], [-4, 1, 1, 0, 3, 0, -4, -1, 4, 4, -2, -1, 0, 3], [2, -1, -2, -3, 3, -2, -4, -2, 2, 2, -2, -1, -1, 0], [-4, 4, 0, -1, -1, -1, -3, -2, 1, 2, -3, 2, 0, 3], [6, -2, 0, -3, 0, 0, 0, -1, 1, -2, 0, -1, -2, -3], [0, 2, 0, -3, 0, 0, 1, -3, 0, 2, -2, 3, -1, -1], [0, 4, -2, -2, 0, 1, -2, -4, 1, 3, -3, 3, 1, 2], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, -2, -1, 1, 0, 1, -2, 1, -1, 0, 1, -3, -1], [2, 0, 0, 3, -1, -1, 1, 3, 1, 0, 1, 0, 1, 2], [5, -3, 1, 3, -1, 1, 2, 3, -2, -3, 2, 1, -1, -1], [5, 1, -5, -4, 2, -3, -1, -2, 0, 0, -2, 1, -3, -1], [6, -3, 0, -2, 0, -1, 2, 1, -1, -4, 1, 1, -2, -1], [4, -2, 1, -2, 2, -2, 1, 2, -3, -2, 1, -3, 1, -4], [-3, -1, -1, 0, 1, -2, -3, -1, 3, 1, 0, -2, -1, -1], [3, 4, -3, -4, 0, -3, -2, -1, -1, -1, -2, 3, -1, 0], [6, -2, -2, -2, 0, 0, 0, 2, -3, -2, 1, 0, 1, -1], [2, 0, -1, 0, -1, 2, 1, -3, 0, 2, -1, 3, -1, 0], [-1, -2, 0, -1, -1, 1, 0, -2, 0, 1, -1, 1, 1, 0], [-3, 2, 0, 1, 2, 1, -2, 0, 5, 3, -3, 0, 0, 3], [-10, 2, -1, -2, 2, -2, -4, -4, 4, 4, -4, 3, 0, 2], [5, 3, 1, -1, 2, 1, -1, -1, 1, 2, -1, 0, 3, 0], [3, -1, 3, 3, 0, 1, 0, 4, 1, 1, 1, -3, 3, 0], [6, -3, -1, -2, 0, 0, 1, -2, -3, 0, -1, 2, 0, -2], [-5, 2, 0, 4, -2, 2, 2, 0, 3, 3, -2, 2, -1, 3], [1, 4, -4, -5, 3, -4, -5, -5, 2, 3, -4, 2, -1, 1], [-3, 4, -6, -2, 0, -1, -2, 0, -1, -1, 0, 3, -2, 3], [-2, 3, 4, 6, -3, 3, 3, 4, -1, 0, 2, -4, 3, 1], [1, 5, 2, 4, -1, 2, -1, 1, -1, 1, 0, 1, 3, 1], [-5, 1, -3, -4, -1, -3, -1, -6, -1, 3, -2, 0, 3, 1], [-3, 2, 1, -3, -1, -1, -1, 0, -1, 1, -1, 0, 0, -4], [6, -2, 4, -1, -3, 3, 1, 3, 1, -2, 2, 1, -3, -3], [3, -5, 1, 3, -2, 0, 4, 3, -4, -5, 3, -2, 4, -2], [6, 2, 2, -1, 0, -1, 2, 0, 2, -2, 0, 0, -1, 0], [11, -2, -2, -5, 4, 1, -1, -1, 2, 1, -2, -2, -2, -2], [-2, -3, 1, -1, -3, 1, 2, 1, -5, -2, 0, -1, 4, -2], [7, 3, -4, -2, -1, 3, -1, -2, -6, 1, -1, 1, 4, -2], [4, 0, -7, -3, 2, -3, -1, -2, -3, -2, -1, 3, 0, 0], [2, 6, -2, 0, -3, -1, 1, -1, -4, 0, -1, 5, 2, 2], [4, -9, 3, 8, -6, 4, 8, 4, -3, -2, 4, -2, 2, -2], [0, -2, 1, -1, -1, 3, -6, -1, -2, 0, 1, 3, 2, -2], [5, 1, -2, 1, -4, 0, 3, 0, -7, -3, 1, 2, 0, 1], [8, -2, -2, 0, -4, 2, 4, -2, -4, -6, 1, 2, -2, 0], [3, 0, -3, -3, -3, 0, 3, -7, -3, 0, 0, 3, -3, -1], [-10, -3, 2, 4, -2, -1, 0, 3, 2, 0, -1, 0, 1, 1], [5, -2, -2, 0, 0, -3, 1, 1, 0, -3, -1, -3, -4, -2], [-3, -3, 3, 2, -1, 3, 0, 4, 4, -3, 1, 0, -2, -1], [14, -4, 1, 2, -3, 4, 9, 1, -4, -8, 4, -1, -1, -2], [-1, 7, -9, -4, 0, -6, -2, -5, 0, 2, -4, 2, -1, 4], [1, -1, 5, 5, 4, 3, 3, 6, 6, -1, 2, -3, -2, 2], [-12, 0, 4, 6, 0, 2, 3, 10, 2, -2, 4, -7, 2, -2], [-2, 1, -9, -7, 1, -7, -6, -5, 1, 2, -6, 3, -2, 2], [-4, 5, -3, -1, 5, -2, -3, 0, 8, 4, -5, -2, -1, 6], [14, -10, 3, -3, -3, 3, 5, 2, -4, -5, 2, 1, -4, -5], [-5, 8, -8, 1, 1, -4, 2, 2, 3, -3, -1, 0, -1, 4], [7, -6, 0, 3, 1, -3, 3, 5, -1, -5, 2, -3, 1, -3], [-12, 4, -1, 2, 1, -1, -7, -1, -1, 6, -1, 1, 3, 2], [-18, 3, 1, 2, -2, 4, 0, -3, 5, 2, -3, 1, 1, 3], [-7, 0, 0, 0, 1, 1, 3, 4, 2, 3, -1, -3, -3, 2], [-4, 0, 1, 7, -1, -2, 2, 6, 0, 0, 5, -2, 0, 5], [5, 0, -1, -1, 3, -3, 1, -3, 1, 1, 2, 3, -1, 1], [3, 3, 2, -1, 2, 1, 0, 2, 3, 3, -3, 2, 1, 3], [-1, -2, -1, 1, -1, -1, -2, -7, 1, 3, -5, 2, 1, -1], [3, 0, 5, 5, -4, 3, 5, 4, -1, -3, 3, -4, -4, 3], [-3, -9, 2, 3, -3, 4, -2, 0, -3, -3, 3, -2, 2, -2], [1, -6, 3, 6, -5, 5, 1, -2, -5, 1, 0, 0, 4, -1], [0, -5, 1, -4, -6, 4, 2, -6, -8, -2, -2, 4, 5, 0], [-2, 1, 4, -1, -2, 4, -1, -5, -2, 4, -2, 2, 2, -1], [6, -7, 4, 6, 2, 1, 0, 6, -2, 0, 4, -7, 2, -4], [6, 3, -5, -8, 1, -1, 1, -2, 1, 2, 1, 4, -3, -1], [3, -5, 2, 2, -5, 5, 4, 1, -1, -4, 1, 3, -3, -3], [7, -7, 0, 1, -7, 5, 8, 0, -9, -6, 5, -3, 2, -4], [-7, 1, 4, 4, -2, 2, 2, 3, 1, -1, 2, -1, -3, 6], [-1, 3, 0, -1, 1, -3, -1, -1, 5, 1, -1, 2, -2, 3], [4, -2, -2, 4, 0, 2, -2, 4, -5, -4, 3, -1, 5, -2], [-7, 5, 0, 5, 2, 3, -2, 2, 5, 9, 1, -2, 5, 6], [1, -6, 0, -1, -1, -1, 0, -1, 5, -3, 0, -5, -3, 1], [-5, 5, -5, -3, 7, -5, -7, -4, 6, 6, -4, 4, -5, 4], [6, -4, 2, -7, -1, 0, -5, -6, -1, 4, -2, -1, 5, -5], [-7, 2, -2, 5, 3, 1, -2, -1, 5, 3, -4, 3, 1, 9], [1, -5, 5, 7, 1, 3, 3, 6, 9, 0, -3, -1, -6, 0], [-4, -1, 1, 6, -1, 1, 2, 2, -1, 0, 6, -1, 1, -1], [2, 0, 0, -5, 2, 1, -5, -10, 3, 6, -2, 1, -4, 2], [-9, 0, 4, 4, 3, 3, -5, 3, 0, -1, 6, -2, 2, 1], [4, 2, -4, 3, 0, 1, 3, 0, 1, -2, 2, 0, 0, 7], [-1, 3, -3, 5, -5, 3, 7, 2, -2, -2, -1, -1, 3, 4], [-8, 3, 4, 1, 2, -1, -7, -5, 6, 12, 1, 2, 3, 7], [0, 8, -1, -1, 3, 0, 2, 1, 3, 0, 0, 3, 2, 2], [-10, -3, 4, 5, -4, 3, 5, 8, -3, 0, 4, -2, 1, 0], [-3, -1, 2, -3, -4, 4, 3, 1, -2, -5, 2, -1, 2, -4], [-3, 8, 3, 4, 1, -4, 1, 7, 7, 5, 2, -1, 0, 5], [7, -1, 2, 7, 4, 1, 1, 7, 3, -4, 3, -1, -3, 2], [8, -9, 2, -2, 5, -4, -2, -3, 1, 0, 3, 0, -1, 2], [9, -1, -3, 5, -8, 1, 7, -1, -6, -5, 2, 2, -3, -2], [16, -6, -1, 0, 1, 3, 1, -2, -4, 0, 5, 4, 1, 1], [-4, 6, -10, -2, 1, -1, 0, -1, 4, -1, -2, 3, -1, 4], [15, 1, -9, 0, 0, -7, 6, -1, -6, -5, 2, 1, -5, -2], [-1, -6, 2, -4, -5, -2, 5, -3, -3, -1, -1, 3, -2, 1], [-5, 0, 5, 3, 3, -2, -1, -1, 0, 0, 4, -1, 0, -1], [6, 2, -6, -5, 2, -1, -2, -2, -1, 6, -3, 1, -4, 2], [3, -7, 8, 1, -2, 0, 4, 5, 0, -5, 4, -5, 4, 0], [8, -1, 7, 1, -3, 4, -4, 0, -2, 0, 1, 0, 5, -3], [0, -1, 8, -1, -2, 0, -8, -1, -2, 4, 1, 1, 4, -3], [-2, 10, -12, -6, 5, -3, -1, -4, 0, 4, -9, 6, -1, 7], [9, -10, -1, -4, -2, -1, 2, 0, -8, -1, 0, 3, 1, -3], [5, -4, 2, 4, -4, 4, 4, -2, -3, -1, 0, 4, 5, 1], [3, -2, -6, -14, 1, -4, -3, -9, 1, 1, -3, 5, -6, -3], [3, 2, -6, 4, -5, 5, -2, -4, -2, 3, 0, 3, 1, -2], [13, -1, -2, 2, -3, 2, 1, 7, -3, -3, 5, -1, 5, -1], [1, -4, 2, 0, -9, 2, 2, -4, -1, 0, -4, 1, 2, -2], [-11, -3, -3, 0, -7, 4, 2, -5, -3, -3, -3, 7, 3, -1], [10, -1, 2, 2, 2, 3, 0, 6, 0, 6, 1, 0, 2, 4], [3, -2, -4, -5, 3, -5, -6, 2, 4, 5, -3, 3, -3, 4], [16, -5, 3, 5, -2, 2, 4, 5, -5, -2, 0, 2, 1, -5], [1, 5, -4, -3, 0, 1, -7, -6, -5, 5, -1, 5, 11, 0], [3, 1, 1, -1, 1, -2, -1, 2, 0, 3, 4, 0, 1, 4], [10, -10, 5, -7, 1, -4, 0, -2, 4, -2, 3, -6, -6, -5], [19, -10, -2, -1, 3, -3, 3, -2, -1, -3, 1, 4, -2, 2], [-3, -1, 0, 7, 2, 2, 7, 9, 1, -5, 1, -2, 3, 4], [5, -7, 2, 0, 4, -1, 1, 7, -3, -9, 7, -9, 4, -4], [1, -2, 5, 4, -1, 1, 0, 2, 8, -1, 2, -4, -5, 4], [22, -14, 5, -6, -4, 4, 11, 4, -4, -12, 1, -3, -2, -9], [-2, -5, 3, -1, 4, 0, 2, -1, 2, 4, 0, 2, 2, -1], [-6, 1, -7, -22, 9, -8, -14, -9, 1, 4, -7, 4, -3, 1], [9, -5, 2, 1, -6, 2, -1, 0, -6, 3, -4, 2, 3, -1], [11, 1, 3, -8, -2, -3, 0, -1, 0, -1, 2, 2, -1, -2], [7, -8, -2, 0, 6, -4, -3, 1, 1, 2, 3, -4, -1, -4], [-12, 1, 0, 3, -3, 3, -2, -6, 3, 5, -6, 2, 4, 1], [-14, 12, -6, -1, 5, -9, -5, -5, 3, 8, -6, -1, 1, 7], [-14, 10, -2, 1, 5, -1, -8, -5, 0, 4, -1, 2, 0, 2], [1, 9, 4, -1, 0, 5, 5, -4, 3, 5, 0, 6, 1, 1], [4, -10, 13, 14, -6, 6, 11, 12, -4, -10, 11, -7, 0, -4], [-2, -5, 4, 1, -8, 3, 3, 0, 1, -2, -3, -2, 3, 0], [9, 0, 3, 1, 1, 1, 2, 5, 1, -1, 1, -7, 1, -1], [13, -14, 7, 1, 0, 4, 1, 4, 1, -1, 5, -9, 1, -5], [1, 4, -1, 3, -1, 1, 3, 1, -7, -1, 1, 0, 9, 2], [-4, 7, 1, -1, 0, -2, -3, -3, 13, 10, -8, -1, -7, 3], [19, -13, 2, 7, -4, 2, 12, 6, -7, -10, 5, -2, 3, -9], [2, 0, -5, 0, 4, -2, 1, 4, 2, -6, 0, 6, -2, 4], [9, -4, -2, -5, 4, -6, -4, 0, -4, -4, 1, -5, -1, -3], [0, -2, -1, 2, -10, 2, 2, -4, -3, -4, -5, 5, 1, -1], [-4, 5, 1, -3, -5, 4, 3, -3, -2, 3, -1, 9, -1, -2], [2, -5, 4, 7, -1, 5, -2, 2, 3, 4, -5, -1, 3, 1], [0, -14, 4, -2, -5, 1, -4, 1, -7, -5, 5, 0, 4, -4], [5, -10, 8, -2, -2, 0, 1, 9, -8, -9, 3, -7, -1, -6], [3, 6, -4, 1, -6, 4, 2, 0, -7, -4, -3, 5, 6, -1], [0, -2, -2, -4, -2, -2, 1, -4, 5, 4, 1, 6, -5, 2], [-5, -2, 5, -6, -5, 1, -6, -7, -2, 2, -4, 8, 7, 3], [10, 2, -2, -8, 1, 1, 8, 6, 1, -2, 4, -3, -6, -2], [6, 3, 5, 5, 2, 1, 2, 8, -5, -4, 5, -8, 1, -4], [-2, -10, 8, 4, 4, 1, 0, 3, 6, -4, 3, -5, -2, -1], [-1, 6, -2, 3, 1, 0, -1, -4, 8, 3, -5, 3, -6, 4], [-26, 15, -4, 2, 5, -3, -5, -5, 7, 12, -4, 1, 2, 12], [-8, 4, 3, 8, 10, -4, -2, 8, 11, 2, 2, -7, -5, 2], [18, 0, -7, -2, 1, -6, 0, -7, -8, 2, -3, 8, 1, 2], [7, -3, -5, -7, -8, 0, -2, -7, -7, -3, -2, 2, -1, -3], [-6, 4, -10, -7, 6, 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-14, -13, 10, 4, 4, -11], [12, -10, -1, 2, -9, 0, 9, -5, -16, -10, 4, 2, 3, -12], [20, -10, 0, -4, 6, 9, 2, 5, -9, -4, 5, -1, -3, -7], [-6, 3, -2, -5, 8, -8, -3, 0, -1, 4, -5, -5, 4, 2], [3, 2, 2, -14, 6, -13, -10, -11, -2, 5, -10, 5, -4, -8], [-13, 2, 5, 12, -9, -1, 5, 10, -1, -1, 4, 5, 2, 0], [-12, 5, 0, 13, -7, 15, 3, 7, -2, -2, 5, -2, 5, -2], [-24, 4, -5, 3, 3, -1, -14, -4, 1, 8, -2, 0, 8, 5], [8, 6, -13, -3, 0, 1, -2, 0, -9, 2, -4, 5, 4, 14], [3, 11, -10, 14, 1, 3, 14, 4, -6, -3, 6, -4, 6, 3], [11, -13, 11, 14, -1, 8, 19, 19, 8, -15, 14, -10, 0, -1], [-7, -8, 2, 1, -8, 8, 10, -2, -9, -12, -4, 6, 2, -5], [-4, 9, -5, 4, 3, 6, 2, 13, 4, 8, 2, 0, 2, 5], [-9, 14, -2, -3, 6, -3, -11, -6, 16, 17, -4, 7, -5, 9], [-4, 1, -3, -7, 5, -1, -6, -1, 2, 10, -7, -4, 5, 6], [-13, 8, 16, 8, 2, 8, 0, 5, 9, 8, -1, -5, -3, 6], [29, -10, 3, -3, 6, 0, 7, 14, -2, -9, -1, -8, -10, -5], [-5, -9, 14, -3, 9, 9, -5, 7, 11, 3, 7, -10, -8, -9], [15, -7, 5, -13, 8, -3, 0, -6, -9, 5, 2, 4, 3, -1], [-6, 17, -7, 2, 1, -4, -11, -5, 7, 2, -18, 6, 3, 10], [32, -15, 17, 0, -8, 5, 3, 1, -4, -10, 7, -2, -7, -3], [-14, -2, 2, 0, -1, 0, -3, 3, -6, 4, 0, 10, 7, 8], [18, -13, 20, 7, 6, -1, 4, 6, 2, -4, 11, -3, 0, -6], [31, 0, 9, 10, -7, 10, 11, 3, -11, -7, 12, 1, 10, -7], [8, -7, -9, -13, 4, -8, 4, -2, 10, -1, -4, 2, -20, 4], [-1, -3, -12, 4, 3, 4, -7, 7, -6, -1, 0, -1, 6, 5], [4, 6, -9, 3, -6, 9, 7, -6, -5, 12, -3, 0, 8, 3], [7, 8, 7, 7, 0, 2, 6, 13, -11, -1, 9, 0, 10, -2], [-15, 22, 3, -14, -2, 0, -15, -11, 2, 8, -11, 15, 2, 10], [-8, 5, 11, 4, -6, 12, 1, -10, 1, 12, -1, 1, 0, 4], [-24, 8, 3, -4, 2, 4, -14, -8, -5, 14, -9, 10, 3, -2], [1, 9, 17, 8, 3, 8, 0, 6, -6, 5, 6, -1, 11, -2], [-23, 2, -8, -8, 6, -5, -6, -3, 15, 5, -11, 0, -13, 4], [4, -3, 14, 14, -11, 0, 12, 14, 8, -9, 6, -4, 2, 3], [12, 4, -2, 7, 5, -3, -2, 1, 3, 4, -5, 2, 7, 6], [-28, 10, -1, 19, -11, 11, 11, 12, -11, 3, 6, 1, 13, 1], [10, -5, -2, -13, 8, -13, 0, 2, 5, -6, 2, 3, -7, -6], [-4, 11, -8, -7, 8, -16, -9, 0, 13, 15, -6, 2, -1, 2], [-22, 5, 7, 10, -5, 9, 2, -2, -2, 10, 3, -2, 12, 1], [20, 1, -4, -13, 2, -4, -3, 1, -17, 0, 6, 2, 5, -1], [21, -1, 8, 11, -6, 1, 8, 12, -15, -7, 13, -14, 7, -5], [25, -25, 13, 11, -3, 17, 10, 14, -7, -12, 16, -5, 4, -2], [-8, 6, 6, -7, 5, -1, 1, -8, 7, 3, 1, -5, -9, -1], [27, -22, -10, 5, 1, 2, 3, -2, -1, -7, 0, 2, -3, -3], [23, -7, 2, 6, 5, -11, -4, 8, -4, 1, 6, -11, 2, -4], [-17, 5, -7, -7, 7, -5, -1, -5, 8, 3, -7, -5, 1, 11], [1, -9, 1, -5, 17, -9, -2, 16, -3, -10, 8, -7, -2, 0], [11, -3, 7, 1, 5, 2, -1, 1, 7, -1, 7, -10, -5, 3], [-11, 0, -18, -4, 3, -12, 0, -5, 10, 5, -5, 10, -7, 3], [8, -8, 5, 3, 5, 0, 3, 12, 8, -10, 1, 0, -8, -6], [5, -19, 8, 0, -4, 5, 6, 9, 0, 5, 6, 3, -1, 2], [-15, 3, 0, -16, 7, 1, -3, -8, -7, 11, 0, 2, 3, 0], [22, -4, 8, 6, -15, 9, 7, -8, -10, 0, -8, 8, 1, 3], [11, -10, -4, -3, 17, -8, -14, 7, 12, 6, -3, -10, 2, -3], [-3, -4, -3, 9, 4, -10, -7, -7, 12, 9, -3, -7, -7, 6], [3, -5, -1, 10, 3, -1, 9, 14, 18, -5, 7, -5, -16, 4], [10, -4, -2, 8, 2, 1, 0, 3, 3, -4, -2, -8, -5, 9], [-10, -8, 20, 12, -13, 17, 4, -2, -3, 8, 0, -11, 8, -6], [22, -7, 3, 12, 0, 10, 7, 18, -11, -2, 13, -16, 8, -6], [-16, 3, 2, -10, -8, 0, -4, -14, -4, 2, -6, 9, 3, 2], [-5, 14, -13, 13, 5, -3, 2, 7, 21, -5, -6, -3, -7, 9], [-4, 10, -5, 3, 2, -10, -5, -7, -1, 4, 3, -1, 10, 5], [-14, -4, 1, -2, -9, 2, -10, -13, 8, 10, -1, 6, 1, 0], [-21, 1, -5, 4, -4, -8, -4, -6, 5, -3, -1, 7, 0, 11], [-16, -2, 3, 7, -18, 14, 6, -5, -8, 2, -9, 3, 9, 5], [-26, 5, 1, 27, 1, 1, 10, 7, 2, 2, 3, -4, 9, 4], [17, -8, 10, 11, 6, 0, -6, 5, 7, 1, 6, -6, -2, 3], [-17, 5, -15, -4, 7, -11, -3, 8, -10, -3, 4, -4, -4, 8], [-7, 9, -15, -6, 4, -9, -10, -9, 8, 3, -3, 11, -3, 4], [5, 0, 7, 13, 3, -2, 15, 11, 9, 0, 6, -7, -1, 11], [-2, 0, 13, -6, -5, 3, -4, -10, 11, 6, -6, 0, -4, 1], [2, -8, 11, -19, 0, -5, -5, 2, 0, 6, 1, -2, -1, -4], [1, -2, 4, 4, -4, 0, 5, 12, 19, -1, -3, -8, -17, -3], [-11, 7, 11, 1, 2, -4, -7, -12, 21, 15, -5, 0, -6, 5], [24, -4, 8, 6, -10, 2, 12, -4, -15, -8, 5, 3, 13, -2], [46, -6, 4, 5, -4, 1, 8, 8, -5, -8, 11, -3, 2, -1], [-18, -7, 12, 10, -1, 1, -4, 4, -4, -16, 1, -1, 10, -1], [-26, -15, 12, 4, -3, 0, 7, 13, 6, 0, 6, -4, -4, -2], [24, -12, 18, 23, -8, 10, 7, 20, -12, -18, 17, -3, 2, -6], [2, 1, -3, -7, -4, -1, 6, -14, 10, -8, -5, 8, -12, 3], [-11, 0, -2, 8, -10, 3, -1, -6, 2, -3, -7, 9, 7, 1], [-9, 6, 5, -4, 3, -2, -5, -5, 1, -3, -3, 9, -2, -5], [-6, 0, -1, -8, 9, -4, -11, -3, 8, 0, -9, 1, -13, 8], [-4, -17, 16, 5, 6, 1, 6, 10, 8, 2, 0, 2, -4, 6], [30, -4, -9, -6, 0, -4, -9, -6, -11, -4, 7, 11, 3, 4], [35, 2, 0, 19, -9, 9, 25, 10, -3, -19, 8, -6, -6, -2], [35, 8, -1, -2, -1, -3, 10, 12, -1, -7, 11, 1, -10, 2], [3, 12, 6, 4, 8, 6, -9, -2, 7, 21, -2, 3, 5, 13], [20, -4, 2, 17, 1, 10, 20, 22, -7, -12, 17, -10, -3, -17], [7, 7, -13, -12, -3, -5, 0, -6, -1, 3, -4, 9, 3, -8], [0, -4, 0, 12, -3, -7, 2, 17, 3, -21, 9, -12, -4, -6], [-6, 0, -6, -4, 10, -3, -8, -8, -9, 6, 1, 4, -6, 2], [22, -2, 8, -1, 9, -5, 0, 13, -7, -2, 8, -11, 1, 2], [-1, 4, 0, -16, 9, 1, -14, -13, 0, 5, -6, -3, -2, -7], [-7, 13, -9, -14, 4, -8, -14, -20, 1, 13, -15, 21, 4, 11], [-4, 15, -16, -10, 4, -8, -16, -12, -6, 6, -15, 16, -3, 14], [15, 8, -9, -11, 4, -5, -2, -16, 11, 9, 1, 9, -6, 5], [-19, 13, 17, 22, -8, 16, 10, 6, -7, -3, 11, -4, 14, -13], [-12, -11, 1, 0, 4, -8, 0, 5, 10, 0, 5, -14, -4, -1], [5, -1, -3, -9, 10, -1, -7, 2, -5, 7, 0, -7, 7, -5]];
    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_p();".
// 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_p(:prec:=14)
    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_p();".
// 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_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8001_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![44, -42, -418, 301, 1134, -875, -1067, 959, 331, -414, -12, 76, -9, -5, 1]>,<5,R![844, -4012, 646, 14710, -5956, -14288, 5316, 5839, -1908, -1158, 335, 110, -29, -4, 1]>],Snew);
    return Vf;
end function;