
// Make newform 8001.2.a.r in Magma, downloaded from the LMFDB on 02 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_r();".
// 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_r();".
// 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 := [4, -148, 416, 1002, -3341, -158, 5821, -2747, -3082, 2282, 565, -712, 9, 98, -13, -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, 0], [0, 1, 0, 0, 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, 0, 0], [3, -5, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [10476522, -23762490, -61371742, 182483997, 112849336, -313228957, -90034741, 203131982, 37846618, -64480493, -8612786, 10901947, 996036, -952649, -45717, 34171], [5521822, 11672715, -63764457, -57366413, 47018671, 65698528, 106565099, -85522528, -102849072, 52754722, 33446774, -14893598, -4685549, 1943846, 241013, -95749], [-10011933, 31200060, 54036313, -188741583, 75977871, 271416573, -316169501, -83578798, 231708473, -19774673, -69724171, 14196617, 9459921, -2447289, -479387, 138556], [-7559716, 194829910, 13012206, -1161023271, 532378722, 1885163751, -1312231707, -982488526, 886177826, 180181899, -256582162, 2433199, 33947762, -4050143, -1689159, 312927], [12678877, -29637740, -178089422, 199086302, 524853251, -387906862, -569728931, 336839837, 284665963, -147009638, -71442151, 33610177, 8756876, -3848034, -416897, 174111], [-3490290, 3158402, 37570814, -543294, -131435121, 9095064, 186611071, -40734444, -108077121, 30643366, 29429019, -9231198, -3782409, 1244303, 185357, -62553], [-11457663, 98969000, 56736448, -736318918, 353563286, 1188437858, -997385891, -546168183, 693632208, 53766867, -204238826, 19868527, 27365426, -5035219, -1375567, 322256], [2433991, 5524390, 185190694, -52865479, -822768247, 201090249, 1148743407, -350064674, -652127926, 217918476, 175479737, -60887074, -22401737, 7892793, 1093459, -387602], [-6777548, 100033165, -15812362, -695428333, 577130586, 1103837898, -1260230866, -449364548, 829385698, 2632702, -238552241, 32948807, 31550366, -6632689, -1572742, 397191], [3556906, 53266175, -120963051, -420987859, 763592393, 646884804, -1357041408, -151345429, 848762304, -92916154, -239584888, 49072551, 31434138, -8006447, -1561421, 443728], [-4486597, 92347345, -26429048, -824771807, 719985124, 1400337842, -1609704994, -592856117, 1070084317, 12346983, -309487504, 40710358, 41075579, -8419656, -2052553, 509784], [-15755608, 109330955, -8611797, -824333673, 794016061, 1336320913, -1711378566, -525465538, 1120057338, -17132488, -321313406, 47185962, 42440381, -9125559, -2114267, 540201]];
    Rf_basisdens := [1, 1, 1, 1, 3260350, 1630175, 1630175, 3260350, 1630175, 326035, 1630175, 1630175, 1630175, 1630175, 1630175, 1630175];
    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, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 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], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 1, 0, 1], [1, 0, 1, 1, 1, 0, -1, 0, -1, 0, 0, 0, 1, 0, -1, 0], [-1, 1, -1, 0, 0, 0, -1, 0, -1, 1, 0, -1, -1, 1, 1, -1], [-1, -1, 0, -1, -1, 0, 2, -1, 2, -1, 0, 1, 0, -1, 0, 1], [-1, -1, -1, 0, 1, 0, 0, 0, 0, 1, -1, 0, 1, -1, 1, 0], [-4, 2, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, 1, 0], [2, 1, 1, 0, 0, 1, 1, 0, 0, -1, 0, 0, 1, 0, -1, 0], [1, 0, 0, 1, -1, -2, 0, 0, 0, 0, 1, 0, 0, 2, -1, -1], [1, 0, 1, 0, 0, 0, -2, 0, 0, 1, 0, 0, 1, 0, 0, -1], [-2, 1, 0, 1, 1, 0, 0, 1, -1, 0, 0, 0, 0, 1, -1, 0], [-3, -1, -1, 0, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, -1], [0, -1, 0, 1, 0, 2, 0, 0, -2, 0, -1, 0, 1, 0, 0, 0], [0, 0, -3, -2, -1, 0, 3, 1, 2, 0, 0, 0, 0, 1, 1, -1], [0, 0, -2, -1, 0, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0], [-4, 1, 1, -1, 0, 0, -1, 0, 2, -1, 0, 1, 0, -2, 0, 1], [1, 0, 0, -1, 0, 0, 2, 0, 2, -1, 0, 1, 0, -1, 1, 0], [-1, 0, -2, -3, 1, 1, 0, -1, 2, -1, 0, 0, 0, -3, 2, 1], [-1, 1, 1, 2, -1, 0, -2, 1, -1, 1, 0, 0, 0, 2, 0, -1], [-1, -1, 0, 0, 1, 0, 0, -1, -1, -1, 0, -1, -1, -2, 2, 0], [3, -2, 0, 1, -2, -1, 0, 0, 0, 0, 0, -1, 0, 1, 2, -2], [0, -1, 0, 0, 0, -2, 0, 0, -1, 0, 1, -2, -1, 1, 0, -1], [-3, 1, -1, 0, 0, 2, 0, 0, 0, 1, -1, 0, 1, 0, 1, -1], [-1, 1, 2, 1, 1, 0, 1, -1, 3, 0, -2, 1, 1, -1, 0, 1], [2, -1, -1, 0, -2, -2, 0, 0, 0, 1, 0, 0, -1, 1, 1, -1], [-1, 2, 3, 0, -1, 0, 0, 1, 0, 0, 0, 1, 1, 1, -2, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -1, -2, -1, -1, 0, 2, 0, 0, 1, 2, -1, -1, 3, -1, -2], [2, 0, 3, 3, 0, -2, 0, 2, -1, 2, 0, -1, 0, 5, -2, -2], [-4, -1, -2, 1, 1, 1, -2, 3, -3, 1, 0, -2, 0, 2, 0, -2], [-3, 0, 2, 1, 0, 0, 1, 0, -1, -1, 1, 0, 2, 2, -3, 0], [-1, 2, 0, 2, -1, -2, 0, 1, -1, 0, 0, 0, 1, 3, -3, 0], [3, -1, -2, -1, 2, 0, -4, 0, 0, 1, 2, -1, -1, -2, 1, 1], [-2, 0, 2, 0, 0, 1, 0, 0, 0, -3, -1, 0, 2, -2, 0, 1], [1, 1, -5, -1, 0, -2, 2, 0, 0, 2, -1, -1, -1, 1, 3, 0], [-2, 1, -2, -3, -2, 0, 0, 2, 1, 1, 2, -1, -2, 1, 3, -1], [1, -3, 0, 2, -1, -4, 1, -1, 1, 0, 1, -1, -2, 3, 0, 0], [-6, -1, 0, -2, 0, 1, 0, 0, 0, -1, -1, 2, 0, -3, 2, 2], [-3, 0, 2, -1, 1, 0, -4, 1, 0, 0, 2, 0, 2, 0, -1, 1], [0, 0, -5, -2, -2, 0, 0, 0, 0, 0, 2, -1, -1, 1, 0, -1], [-7, 5, -2, -2, 1, 0, 0, -1, 0, -1, 2, -1, -1, -1, -1, 1], [-3, -2, -2, -2, 0, 2, 4, 0, 1, 0, 0, 0, 2, 0, 0, 0], [0, -2, 2, -1, -1, 0, 2, -3, 3, -4, 0, 2, 0, -1, -1, 3], [-3, -2, 1, 1, 2, -1, 0, 0, -2, 0, -2, 0, 0, 0, -2, 1], [-7, 2, 3, 2, -1, -2, 0, -1, 1, 0, 0, 1, 0, 2, -1, -1], [-3, 2, 0, -1, -1, -1, 0, -1, 3, -1, 2, 0, 0, -2, -1, 2], [-2, 1, 3, 2, 0, 0, 2, -1, 2, 0, -2, 1, 2, -3, -1, 3], [0, 0, -3, -4, -3, 0, 2, 0, 3, -1, 2, 0, -2, -2, 0, 0], [1, 0, 3, 4, 3, -1, -4, -1, -1, 0, 1, 0, 0, 1, 0, 1], [3, 5, 1, 0, 2, 2, -4, 1, -4, 0, 3, -1, 1, 3, -3, -1], [2, -3, 0, -1, 1, 2, 2, -1, 2, 0, -2, 0, 3, -5, 0, 1], [-4, 0, 1, 3, 1, 2, -3, -1, -3, -1, -2, -1, 3, -2, 0, 3], [-3, 0, 0, -3, 0, 0, 5, 1, 3, 0, -1, 1, -2, 1, 1, 0], [-5, 0, 3, 0, 2, 4, -2, -2, -2, 1, -2, 0, 4, -4, 0, 1], [3, -1, -3, 2, 1, 0, 0, -1, -1, 1, -2, 0, -2, -2, 0, -1], [-2, -1, -1, -2, -3, 0, 2, 2, 1, 0, 0, -1, -2, 0, -1, -4], [4, -5, 1, 3, -1, -1, -2, -1, -4, 1, -3, -1, -1, 2, 2, -1], [-6, 6, -2, 0, 1, -2, 2, -1, -1, -3, 4, 1, -3, 3, -3, 4], [-3, -4, 7, 3, 0, 3, 2, 0, 2, -2, -2, 2, 7, -3, -2, -1], [-8, -1, -3, -1, 3, 2, 0, -1, -4, 2, -4, -1, 0, -3, 4, -2], [0, -1, 0, 1, 0, -1, -2, 0, -5, 2, -3, 1, 2, 1, 1, 0], [-6, 3, -3, 0, 0, 0, -2, -2, -4, -1, 0, 0, -2, 3, 1, -1], [1, -2, 4, 1, -2, -2, -2, 2, 3, 0, 4, -2, -1, 2, -1, -4], [8, -6, 0, -1, -2, 0, -2, 2, -2, 3, 1, -1, 1, 4, -1, -4], [-2, -2, -2, -3, 0, 0, 4, 1, 4, -1, -1, 1, -3, -1, 2, 1], [0, -2, 6, -1, 1, -1, 0, 1, 2, -2, 1, 3, 1, -3, -3, 2], [-7, -6, 1, -4, 1, 2, 1, -3, 1, 1, -1, 1, 2, -4, 3, -1], [-6, 1, -1, 0, 0, 0, 0, 1, -2, 2, 0, 1, 1, 2, -1, -3], [-4, -6, -4, -2, 0, 2, 5, -2, 0, 0, 1, -1, 0, -2, 5, -1], [8, 3, -2, 1, 0, 0, -2, 0, 1, -3, 4, -3, -2, -2, -1, 3], [-3, 1, -3, -4, 1, -2, 4, -1, -1, -1, 2, 0, -6, 2, 0, 1], [0, -4, 1, 1, -1, -2, 4, 2, 0, 0, -2, -1, 3, 4, -3, -4], [3, -4, 7, 3, -3, -2, 0, 3, 4, -2, 5, 1, 2, 5, -6, -1], [0, 1, -2, -4, 2, 0, -1, 0, -1, 0, 2, -1, -1, -1, 1, -2], [-3, 5, 4, 0, -1, 0, -2, 0, 1, -1, 2, 1, -1, -2, -3, 1], [1, 4, 1, 5, 6, 0, -6, 0, -6, 2, -3, -1, 2, -3, 3, 3], [4, 3, -6, 1, 0, 0, 1, -2, 0, 0, 0, -1, -1, -1, 1, 3], [-6, -1, -3, -2, 3, 2, 2, -1, 1, -1, -2, 2, -3, -4, 3, 2], [3, -2, -4, -1, 1, 2, -2, -1, 3, -1, 2, -3, -2, -2, 1, 1], [0, -3, 2, -3, 2, 6, -1, -2, 1, 0, -3, 1, 4, -9, 1, -1], [-6, 4, 3, 0, -1, -1, -2, 1, 3, 0, 1, -2, -2, -1, 1, 2], [1, -11, 6, 0, -2, 0, 3, 2, 7, 0, -2, -1, 3, -2, 2, -3], [6, -4, 0, 1, 0, 0, 2, 0, -4, 1, 0, -1, 2, 4, -4, -1], [-8, 5, -5, -3, -1, 3, -2, 1, -6, 3, -2, 1, -6, 0, 7, 0], [2, -4, 0, -3, -1, 0, 2, 1, 4, 0, 2, 2, -2, 0, 4, 2], [3, 0, -2, 1, -1, -2, -3, 1, -2, 0, -1, -3, -2, 3, -2, -2], [-1, -2, -3, -2, 1, 0, -4, 1, -5, 2, 0, 1, -4, 2, 3, 1], [0, -3, 5, 6, 1, -2, 0, 4, -2, 3, -2, 0, 0, 11, -3, -4], [-4, -1, 6, 0, 3, 4, -2, -1, 2, -3, -3, 3, 2, -8, 1, 1], [-8, 3, -1, -2, 2, 0, -2, 0, 0, 2, -2, -1, 1, -4, 2, -2], [-5, 1, 5, 3, 0, 0, 0, 0, 0, 0, -1, 3, 1, 1, -5, 0], [-1, 2, -1, -1, 1, 2, -4, 2, -2, -1, 1, -4, 0, -1, 0, -3], [3, 1, 2, 2, 0, -4, -2, -1, 1, 1, 1, 1, -5, 2, -1, 6], [3, -2, 0, -2, -1, -4, 3, 1, 2, 1, 2, 2, -2, 5, 1, 2], [-2, 2, 2, -5, -1, 0, 4, -1, 7, -1, 2, 3, -3, -5, -2, 4], [-4, -11, 8, 1, 0, 2, -2, -2, 1, -2, -2, 2, 5, -7, -2, 0], [-1, -1, -4, -5, -1, 0, 8, -1, 7, -1, -1, 3, 1, -4, 2, 3], [0, -3, 2, 6, 1, 0, -4, -1, -1, -2, 0, -2, 1, 2, -4, 2], [-8, 3, 2, 3, -6, 0, 2, 2, 1, -2, 4, -2, -3, 7, -2, 0], [-7, 5, -4, -2, -4, 2, 2, 2, 0, 0, 1, -2, -3, 3, 2, -1], [-1, -3, 2, 3, -3, 0, -2, -1, -2, -1, -2, 1, 3, 1, 0, 1], [3, 4, -2, -1, 2, -2, -5, 0, -6, 0, 6, -3, -6, 1, -6, 3], [2, -4, -3, 4, 0, 0, 1, -1, 1, 1, -2, 1, 3, 1, -1, -2], [-5, 3, -5, 2, 3, -2, 0, 1, -4, 4, -3, -3, -5, 1, 3, 0], [4, -7, -5, 2, -1, 2, 4, 0, -3, 1, -2, 0, 0, 2, 4, 3], [-9, 4, 2, 0, 2, 1, -6, 4, 1, 2, 2, 1, -2, 3, 0, -3], [0, 5, 3, 5, 1, 2, 1, 1, 2, -2, 1, 0, 3, 2, -3, 1], [10, 0, -3, 0, 1, -2, 0, 3, -2, 4, 3, -5, -1, 7, -2, -6], [-4, 2, 5, -5, 0, 0, -4, -1, 1, 2, 0, 1, -1, -4, 3, 1], [-10, 1, -5, -7, -2, 3, -2, 0, 3, 1, 1, 1, -1, -3, 2, 0], [-2, -2, 4, 3, 0, 2, -6, -2, -5, 1, 2, 1, 1, 1, -3, 2], [-2, -1, 6, 7, 0, 2, 0, 1, -4, -2, 1, 0, 2, 0, -3, 3], [-8, -2, -6, -2, 1, 2, 0, -3, 2, -1, 0, -1, 1, -7, 4, 1], [4, -2, 0, 1, 1, 4, 10, -1, 4, -1, -6, 3, 9, 0, -3, 1], [3, 1, -1, 1, -1, -4, -2, 1, -6, -1, 5, -2, -8, 11, -1, 1], [2, 0, 0, 0, -2, -4, 0, 4, 3, 0, 3, -2, 1, 3, -2, -2], [-2, -6, 0, -1, 0, 0, 4, 0, -1, -1, -3, 3, -1, -6, 0, 1], [12, -6, 6, 0, -2, -3, -1, -2, 3, -1, -1, 1, 4, 0, -1, 2], [-16, 0, -4, -2, 2, 0, 2, -1, -2, -2, 0, 0, -7, 2, 3, 1], [-8, -4, -4, 2, 3, 0, -4, 1, -8, 3, -3, -1, 0, 1, 1, -4], [-4, -7, -1, -2, -3, -1, 2, -3, 1, -2, 1, 0, 1, 2, -1, 1], [-9, 2, 8, 1, 1, 0, 0, -1, 4, -2, 2, 6, 2, -3, -3, 3], [-4, 0, -6, 0, -2, 1, 4, -2, -3, -3, 1, 0, -2, 6, -1, 0], [-4, 5, 4, -1, -3, 2, -1, 2, -2, -1, 5, 0, -2, 2, -3, -2], [-7, 0, -1, -1, -2, -3, -4, 0, 0, 2, 4, -4, -6, 6, 4, -1], [10, 5, -6, 1, 2, 4, 6, 0, -4, 0, -3, 0, 0, 3, 2, -4], [-1, 3, -2, -2, 2, -1, 0, 0, 4, 1, -2, 0, 0, -2, 0, 0], [-6, 5, 6, -1, 2, 0, -4, -1, 6, -1, 1, 1, 3, -5, -2, 0], [-1, -7, -4, -3, -6, 2, 6, 2, 1, 1, 0, -3, -1, 4, 4, -6], [0, 1, 2, 5, 3, 3, -5, -1, -3, -3, -3, 1, 1, -4, 1, 3], [-6, 0, -1, 0, -2, 0, 2, 0, -1, 5, -3, 2, 4, 0, 5, -4], [7, -1, -1, 5, -3, -3, 0, 1, -2, -2, -1, 0, 0, -1, -2, 1], [-15, -7, 3, -3, -3, -2, 3, 0, 3, -3, 2, 1, -3, -2, 4, 0], [6, -9, 0, 6, 3, 0, -2, 3, -4, 4, -6, -6, 3, -1, 3, -7], [0, 7, -3, -5, 0, 0, 4, 4, 3, 1, 4, 0, -1, 4, -3, -4], [-10, -6, -6, -3, 2, 2, -1, 0, 1, 4, -5, -1, -1, -2, 11, -2], [-5, -4, 9, 3, 6, 6, -2, 0, 1, -3, -7, 2, 10, -7, -1, 1], [-1, -3, -4, -1, -6, -2, 2, 1, 3, 1, 4, -1, -4, 4, 3, 1], [-2, 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-8, -2, 0, 1, 1, -2, -3, 1], [14, 3, 3, -1, -1, 0, -2, -1, 2, -4, 3, 3, 5, -1, -13, 7], [-8, -10, -3, -3, 3, -2, 2, -3, 0, 0, -6, 3, -1, -13, 4, -3], [-2, 7, -5, -2, 1, 2, -2, -1, -6, 0, -1, -7, -2, -2, 2, -11], [14, -2, -3, 6, 1, 4, -4, 0, -7, 3, -2, -2, 6, 4, -2, 0], [13, -12, -4, -2, -2, -4, 7, 1, 2, -4, 5, 1, 0, 1, -7, -1], [-9, 10, -4, -5, 4, 5, -7, -4, -7, -1, -6, 3, 2, -6, 2, 4], [6, 5, 3, 5, -5, 2, 6, 5, 5, 0, -1, 1, 8, 8, -6, -6], [-13, 0, -8, -3, 6, -7, -8, 0, 0, 2, -2, -1, -6, -5, 2, 2], [4, -10, -2, 0, -8, -2, 12, 4, 6, -2, 2, 0, 4, 10, -2, -6], [-8, -10, 10, 6, -2, 0, 4, 0, 3, -3, -7, 3, 9, -9, -3, -3], [3, -10, 10, 5, 3, -6, 0, -1, -3, 0, -3, 0, 3, 0, -3, 3], [18, -13, -3, 3, -3, -5, 2, 3, -7, 8, -2, -6, 8, 9, -1, -14], [-6, -1, 6, 4, -3, 5, -2, 3, 0, 0, 0, 1, 2, 0, -1, -8], [5, 3, -5, 4, 7, 6, 10, -1, -1, -3, -11, 2, 12, -1, -3, 0], [23, -1, 9, 10, 4, -5, -2, -6, -5, -2, 3, 2, 3, 6, -7, 4], [1, 0, 3, 2, -1, -8, -10, 1, -5, 4, 4, -7, 0, 10, -3, -5], [13, 6, 3, -1, 4, -4, 2, 2, -2, 0, 9, -3, -2, 4, -7, 3], [8, 3, 1, 5, 10, -4, -10, 0, -14, 9, 1, -4, -5, 6, -5, -6], [-2, 9, 6, -2, -1, -4, -2, 1, 8, -6, 7, 0, -5, -7, -4, 14], [-8, 8, -2, -3, 8, 8, -1, -5, 3, -6, 1, 1, -2, -11, 6, 3], [8, 3, -3, 2, -3, 2, -2, 1, -7, 5, -2, -4, -5, 3, 3, -4], [3, -11, -12, -4, -4, -4, 4, -2, 7, 5, -3, -3, 0, 4, 6, -11], [4, 9, 2, 1, -5, 0, 2, 1, 3, -2, 1, -2, -2, 3, -6, -5], [9, 1, 12, -2, 0, -2, 2, -6, 8, -13, 2, 5, -3, -8, -2, 7], [-1, -3, -5, 2, -2, 0, 2, -2, -3, 5, -11, -2, 7, -3, 6, -4], [9, 4, 6, 7, -1, -1, -6, 5, -4, 1, 3, -6, 1, 13, -5, -6], [16, -6, 2, -5, -6, -4, 4, -4, 9, -3, -2, 5, -5, -2, 5, 8], [-3, 7, 3, 0, -5, -8, 2, -2, 4, -5, 1, 6, -3, -1, -1, 0], [3, 6, -2, -9, 4, 0, -1, 0, 7, 2, -3, -1, 3, -3, -2, -1], [3, 4, 9, 11, 4, 0, -5, 6, -8, 3, -4, -1, 3, 8, 1, 3], [11, -12, -7, 2, -8, -6, 1, 2, 5, 4, 2, -4, -1, 2, 4, -5], [23, -8, 5, -3, 9, 6, 0, -5, 1, 3, -11, 4, 7, -14, -2, -3], [0, 11, -2, -1, 1, 2, 5, 3, -2, 0, -1, 5, -1, 6, -7, 0], [-2, 3, 9, -2, 7, -1, 4, -3, 11, -6, 3, 6, 4, -13, -11, 9], [19, -2, 5, -1, -8, -4, 1, -1, 4, -1, 10, -1, -5, 9, -2, 6], [4, 2, 7, -4, 3, 2, 2, -4, 4, -6, 0, -1, 7, -4, -4, 2], [-7, -3, 9, 4, 3, 10, 3, 5, -4, 3, -10, -1, 9, 2, 1, -12], [8, 13, 3, 0, 2, -6, -3, 8, -5, 3, 6, -1, -6, 16, -10, 2], [-19, 4, -11, -6, 2, 8, -4, 4, -5, 8, -7, -1, 5, -4, 9, -8], [-26, -5, 2, -5, -5, 3, 13, 1, 9, -8, -1, -2, -4, 3, 2, -2], [0, 0, 12, 8, 0, -4, -7, -4, 1, -4, 2, 5, 2, -8, -1, 9], [-1, -10, 9, 0, 3, 10, -2, 5, 1, -2, -4, 1, 12, -14, -1, -5], [6, 1, 9, 2, 0, 4, 6, 0, -2, -5, -4, 0, 9, -1, -9, -5], [8, -2, 4, -3, -10, 0, 5, 2, 0, -1, 0, 2, 7, -1, 2, -3], [-3, 1, -14, -10, -4, 4, -4, -2, 4, 3, 6, -5, -6, -5, 12, -3], [-20, 6, 1, -7, 0, 6, -4, 2, 3, 0, -4, -1, 5, -8, 8, -6], [4, -15, 7, 5, -2, -4, 3, 2, -5, 2, -1, 0, 1, 6, -3, -4], [-9, -20, 2, 6, -2, 2, -4, 0, -11, 1, -5, -3, 5, -3, 2, -8], [-12, 6, 1, -5, -1, 2, 2, -1, -4, -4, 3, 1, -2, 10, -3, -2], [5, -18, -9, -8, -4, 4, 12, -6, 10, 1, -10, 6, -3, -12, 10, -3], [5, -8, 5, 6, 3, -8, -10, 1, -7, 8, 0, -5, 0, 6, -3, -7], [0, 10, -13, 0, -3, 0, -2, -1, 3, -1, 6, -6, -14, 14, 7, 3], [-13, 8, -1, 4, 3, 0, -2, 5, -2, 5, -9, -2, 7, 5, -2, -4], [23, -16, -11, -8, -5, -8, 6, -1, 3, -1, 4, -2, -7, -1, 2, -4], [-1, 11, 5, 5, 9, 4, -5, -3, 5, -3, 2, 7, -2, -6, -1, 9], [12, 1, -2, -8, -4, -2, 6, -2, 4, -8, 6, 2, -9, -9, -2, 9], [-4, 7, -3, 7, 12, 0, -6, 2, -11, 1, -2, 0, 2, 3, -3, -2], [-14, 0, 2, -1, 0, 0, -1, 0, -5, 4, 3, -1, 0, 14, 5, -2], [-4, 8, -9, -15, -1, 0, -4, 5, 1, 7, 7, 0, -8, 4, 1, -8], [-15, 0, -4, -4, -4, -2, 6, 2, 0, 6, -2, 6, -7, 9, 9, 1], [7, 5, 4, 1, -5, -4, 0, -1, 2, -6, 8, -2, 0, 0, -9, 6], [14, 5, -3, -10, 4, 2, 8, -2, -1, -2, 11, -1, -4, 1, -10, 0], [4, -3, 0, -3, -1, 4, -1, -1, -4, 5, -3, 4, 5, 1, -8, 5], [-5, -1, 1, 3, 1, 2, -4, 3, -2, -4, 0, 3, -3, -3, 5, 9], [-26, 1, 5, -1, 3, 12, 12, -1, 15, -8, -12, 11, 9, -19, 7, 9], [-1, 7, 3, 8, 4, 2, -4, 1, -3, 2, 3, 7, 1, -2, -4, 6], [1, -5, -9, -12, -13, -2, -2, 3, 2, 4, 7, -3, -5, 7, 7, -4], [-8, -7, -19, -8, 7, 4, 3, 1, -1, 4, -11, 4, 1, -7, 13, -4], [-2, 0, -6, -2, -9, -12, -4, -2, 4, 1, 5, -3, -3, 8, 0, -3], [-8, 4, 5, -5, -1, -3, -2, -1, 10, 0, -1, 4, 3, -10, -1, 5], [-11, -15, 6, -3, -1, 0, -4, -3, -5, 1, 3, -5, 3, -2, 4, -5], [-2, -9, -2, -10, 5, 2, 4, -1, 9, 1, -4, -1, -2, -3, 7, 1], [20, -14, -6, 7, 0, -4, 2, 0, -9, 7, -8, -5, 9, 11, 3, -14], [-9, 5, -2, 2, -6, -8, 14, -2, 8, -4, -2, 2, 2, 5, -5, 6], [-22, 0, 10, 3, 6, 0, -6, -2, -7, -2, -6, 2, -2, -3, 0, 7], [-21, 11, 6, 7, -2, -2, 4, -4, 1, -3, -1, 7, -4, 0, -11, 9], [-9, 1, 2, 6, 2, -4, -2, 0, -4, -1, 5, 1, 3, -4, -3, 4], [24, 3, 10, 3, 1, 2, -4, 7, -11, -2, 3, -2, 1, 5, -8, -5], [1, -4, -9, -4, 3, -8, 0, 1, -1, 5, 1, 0, -13, -2, 5, 3], [12, -3, 14, -3, 2, 2, 8, -4, 12, -4, -1, 8, 13, -9, -10, 10], [5, 0, 3, -2, -3, 5, 2, -11, -4, -3, 3, 1, -5, -5, 6, 5], [-5, 7, 9, -5, 0, 2, 8, -4, 12, -10, 2, 7, -4, -3, 2, 13], [-7, 7, -16, -5, 0, 4, 6, 1, 1, 1, 4, 1, -2, -4, 1, -3], [1, 9, 3, 2, 12, 2, 0, 0, 5, 0, -1, 5, 7, -8, -5, 7], [6, -2, 3, 5, -2, 3, -6, 0, -5, 1, -2, -1, 7, -2, -3, -8], [-6, 0, -6, 6, 1, -6, -12, -1, -14, -1, -1, -5, -3, -3, -3, -4], [-7, -7, -3, -2, 6, 4, 0, -5, 0, 2, -4, 1, 2, -8, 8, -4], [-4, -1, -1, -2, 2, 2, 2, -6, 7, -2, -5, 1, 2, -9, 6, 8], [7, 5, 13, 12, 8, 6, -12, 2, -3, 3, -9, -2, 10, -4, 0, 0], [5, 1, -7, 0, -3, -6, -2, 2, 0, 3, -1, -4, -6, 6, -3, -2], [-11, 5, 2, -2, -5, 3, 12, -3, 11, -6, -1, 7, 1, 7, -2, 11], [-27, 1, -1, -5, 0, 7, 6, -2, 4, -4, -1, 0, 5, -8, 5, -7], [10, -11, -13, -8, -6, -8, 4, -2, 12, -1, 6, -2, -9, 10, 5, 3], [8, 12, 5, -1, -5, -4, -6, 0, 3, -3, 14, 0, -8, 5, 3, 10], [2, 6, 0, -13, -5, 2, 8, -3, 4, 0, 2, 0, -8, 7, 0, 4], [8, 8, -1, -9, 1, 0, 3, -3, 11, -1, -8, 9, 3, -10, 10, 3], [-4, -2, -17, -6, 1, 4, -4, -2, -5, 10, -4, -1, -5, -3, 13, 1], [0, 1, 9, 5, -1, 2, -4, 1, -4, 6, -2, -3, 4, 5, -2, -8], [6, 6, 1, 5, -3, 4, 6, 5, 5, -1, 3, -2, 0, 13, -3, 4], [2, 3, 6, -3, 0, -2, -8, 0, 6, -2, 6, 0, -2, -8, -7, -1], [-10, -4, 2, -9, 1, 0, 2, -4, -5, -1, -1, 1, 1, 0, -10, 0], [-26, 8, -8, 6, 0, -1, -2, -4, -11, 9, -10, 0, -2, 3, 8, -1], [-9, 8, 7, 4, 4, 7, 4, 0, 4, -6, -7, 6, 6, -5, 2, 6], [2, 4, 2, 7, 2, 6, -9, 2, 1, 3, 1, 0, 6, -6, 4, -1], [-5, 2, -6, -9, -11, -2, 6, 1, 11, 5, 3, 1, 0, -1, 4, -4], [-37, -2, 2, -2, 0, 4, 10, 0, 4, -1, -6, -3, 6, 3, 8, -2], [5, -7, -5, -5, -5, -4, -2, 0, -3, 9, 7, -6, -8, 10, -1, -11], [-22, 20, 3, 0, 3, -1, -8, 5, -5, 0, 9, -2, -10, 15, -9, 4], [-1, -1, -14, -3, 1, -6, -8, -2, -10, 11, -2, -7, -9, -2, 15, -4], [12, -3, -11, -7, 3, 0, 0, -3, 6, 7, 1, -4, 0, 0, -2, -8], [0, 3, -5, -6, -2, 0, 1, 6, 0, 2, 9, -2, -5, 19, -5, -9], [-3, 1, -4, -4, 1, -2, -6, -3, 11, -4, 7, 2, -6, -5, -4, 10], [-1, 12, -3, 1, 2, 0, 6, 2, 5, -2, 6, 3, -6, -1, -5, 3], [5, 1, 6, -1, 1, 4, 4, -6, 6, -3, -4, 3, 7, -2, -9, 0], [-8, -3, -5, -1, -1, 1, -2, -5, -1, 3, 2, 1, -1, 2, 4, -5], [4, 6, -4, 5, -4, -6, 4, 1, 2, 3, 8, -3, -3, 22, -7, -8], [-18, 4, 11, 5, -3, -8, -2, 3, 4, -1, 2, -2, -4, -2, -3, 2], [3, 2, 1, -11, 0, 0, 0, 3, 10, -8, 6, 3, 2, 4, -7, 5], [3, 18, -4, -4, -3, -4, -6, 0, -3, 1, 0, 3, -3, -2, 3, 2], [-29, 11, -2, -2, 2, -1, 4, -2, 2, -8, 6, 3, -3, -9, 1, 13], [-11, 14, -19, -11, 1, 4, -6, -1, -4, 9, 6, -6, -9, 1, 8, 1], [-17, 2, -5, -4, -7, -3, 6, -1, 16, -1, 0, 1, 1, 6, 3, 0], [11, -8, 14, -4, 6, 4, 4, -2, 20, -8, -6, 10, 13, -26, -3, 7], [1, -17, 6, -4, -10, -4, 8, -2, 8, -5, -1, 3, -2, 0, -3, 0], [-12, -7, 0, -2, 12, 11, 8, -2, 4, -4, -13, 3, 9, -20, 3, 7], [3, 8, 6, 5, 2, 0, 3, 4, -5, -1, 5, 6, -7, 7, -11, 8], [9, 1, 0, -8, -5, -2, -6, -1, 5, -1, 2, 1, -3, -17, -4, 6], [28, 7, 19, 9, -1, -13, -1, -1, 8, -6, 16, 5, 3, 5, -20, 11], [-17, -9, 0, -10, -1, 2, 5, -3, 10, 0, -2, 5, -2, -5, -2, 0], [1, 10, -6, -7, 7, 8, -7, -3, -6, 3, -5, 2, -6, -9, 3, 7], [-10, 7, 10, 3, -1, 6, 8, -7, 0, -10, 4, 2, -2, 1, -4, 9], [-14, 6, -14, -5, -1, -2, -5, -3, -3, 1, 0, -4, -7, -10, 8, 0], [-11, 0, 3, 1, 4, 2, -2, -6, -7, 7, -6, 0, 1, -2, 6, -4], [-40, 4, -13, -13, -7, -4, -9, -3, 3, 6, 2, -2, -5, -1, 3, -2], [-9, -3, 12, 4, 1, -6, 10, 2, 1, 4, -3, 4, 3, 11, 1, -3], [-2, 12, 16, 11, -4, -6, -1, 4, 6, -12, 12, 3, 3, 7, -13, 2], [-1, 9, 17, 2, 4, 2, 9, 2, 2, -9, -3, -1, 5, -1, -5, -1], [19, 11, -14, -3, -11, -4, 11, 3, 12, 4, 5, -1, -10, 19, 2, -5], [13, 5, 11, 10, -5, -2, 2, 2, -2, -2, 3, -1, 2, 16, -18, -3], [21, 8, -1, 6, 6, 2, -4, 2, -15, 5, -6, -2, 7, 4, -5, -6], [-9, 12, 13, -1, 2, 2, 0, -4, 9, -6, -6, 9, 2, -8, -1, 5], [8, 8, 6, 10, -4, -2, -18, 6, -12, 12, -5, -6, 3, 15, -1, -11], [-9, -10, -5, 3, 2, -4, -1, 4, 3, -4, 3, -6, 0, -9, -4, -4], [25, -2, 1, 4, 6, -4, 0, 0, -2, -1, -6, 4, -1, 2, -2, -1], [6, 1, 14, 15, 1, -4, 0, 0, 5, -1, -11, 5, 9, -4, -8, 9], [-3, -1, -7, -12, -2, 4, 7, 0, 1, -3, 0, 1, -13, 9, 1, -3], [-5, 3, 6, -5, 3, -2, -3, 1, 10, -1, -1, 10, -5, -4, 7, 6], [35, -14, 16, 7, 4, -3, 0, 4, 4, 0, -4, 1, 13, 4, -12, 0], [-11, 0, 17, 5, -3, -4, -14, 0, 3, 4, -1, 1, 1, -5, 0, -1], [8, -13, 14, -2, -8, -12, 4, 6, 9, -5, 6, 5, 1, 5, -6, -5], [7, 1, 7, -12, -8, -2, -2, -2, 12, -6, -2, 5, 3, -10, 5, -1], [-24, -6, 7, 6, 11, 6, 2, -9, -7, 1, -8, 6, 6, -1, 5, 7], [-16, 8, 5, 1, 2, 6, -8, 4, -1, -1, 2, 2, 6, -3, -1, -1], [-23, 7, -11, -3, -4, 2, -2, -5, -2, 0, 3, -7, -8, -7, 12, -1], [-5, -11, 5, 8, 2, 7, 0, 0, -1, 1, -6, -3, 4, -6, -1, -6], [4, -8, -11, -1, -3, -2, -2, 5, -5, 13, -10, 0, 0, 8, 4, -11], [-1, 6, 8, 2, 9, 2, -4, 1, -1, -1, 6, -3, 6, 2, -10, -5], [3, 3, -3, 2, 12, 6, -6, -2, -5, -6, -3, 3, 0, -6, -5, 13], [4, 11, 1, 7, -6, -14, -4, 4, -7, -3, 12, -2, -17, 12, -5, 4], [18, 4, 2, -7, -3, 0, 4, 1, 9, 1, 11, -1, -1, 6, -1, -1], [8, -1, 7, 7, -2, -2, 1, 3, 2, 0, -4, 2, 7, -1, 6, -1], [-22, 0, -9, -2, 1, -4, 3, -3, -1, 3, -11, 3, -4, -6, 8, -4], [-9, -12, -2, 8, 4, -4, -6, -1, -10, 7, 1, -7, -1, 2, -4, -10], [-4, -9, 6, 8, -2, -6, -4, 2, -11, 6, -12, -6, 8, 6, -5, -14], [0, -8, -17, -4, -1, 6, 4, 2, -7, 7, -12, -4, 0, 4, 10, -16], [2, -11, 0, 5, -1, 2, -2, -3, -10, -3, 9, -7, -1, 6, -4, -1], [-4, 5, -4, 3, -1, -4, -4, 7, 1, 8, -1, -6, -7, 3, 10, -7]];
    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_r();".
// 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_r(:prec:=16)
    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_r();".
// 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_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8001_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![4, 148, 416, -1002, -3341, 158, 5821, 2747, -3082, -2282, 565, 712, 9, -98, -13, 5, 1]>,<5,R![-27136, 208768, -344128, -359328, 542544, 232104, -316420, -73336, 91364, 12600, -14371, -1185, 1249, 56, -56, -1, 1]>],Snew);
    return Vf;
end function;