
// Make newform 8046.2.a.n in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8046_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_8046_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_8046_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 := [-3, -594, -1607, 345, 3573, 1551, -1561, -684, 334, 82, -31, -3, 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], [-135751602546, -403274368788, 294511464920, 994832840887, -38007995419, -465585432656, 50310595815, 62879646690, -8579109025, -2862765831, 373555153, 24748733], [-100388062134, -184654740606, 403759068040, 495900208592, -377751239522, -208981623823, 175851726345, 3959704842, -22317146501, 2550850410, 838314338, -142183724], [65302848768, 651960038514, 745303063093, -1150914398347, -1672929464267, 479743706735, 697617356013, -164381452443, -83789564729, 19932549972, 3071881331, -743676686], [915421136, -177941248499, -193999453699, 427946138875, 528630702028, -196533632656, -226011640501, 61692870961, 27363214940, -7145245871, -1008357826, 260555422], [-29055232476, -830466305577, -899515205951, 1819129337198, 1806674571358, -843237849712, -686060598954, 218268434445, 79587640096, -22855402146, -2873219941, 791457988], [-7334114235, -857897373063, -954029651272, 1735102707136, 1958545237151, -806208285203, -779842057185, 230188534851, 92407418951, -25430053989, -3376570442, 905897063], [44724727668, 293198015341, 140774506663, -689122894818, -556795852562, 313671881091, 238485257763, -82150526984, -28833755646, 8694135210, 1063333243, -303077175], [17837198367, 318441901306, 297129010822, -783228359697, -709008838650, 378855759913, 271887620614, -94978496783, -31578991963, 9678326681, 1140734420, -329344029], [-27070193104, 322371833162, 389355266298, -702253132145, -756307087494, 328556990398, 299777725695, -91893899224, -35438802905, 10017640191, 1292153510, -353735657], [-1717939244, -384310270718, -700613773249, 836181281764, 1522325564529, -374724551769, -647726374616, 146244224743, 78645513257, -18475689520, -2905422747, 701406468]];
    Rf_basisdens := [1, 1, 24466366011, 24466366011, 24466366011, 8155455337, 24466366011, 24466366011, 8155455337, 8155455337, 8155455337, 8155455337];
    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_8046_a();"
function MakeCharacter_8046_a()
    N := 8046;
    order := 1;
    char_gens := [299, 3727];
    v := [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_8046_a_Hecke(Kf)
    return MakeCharacter_8046_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, 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, -1, 0, 0], [-1, 0, 0, 0, 0, 0, -1, 1, 0, 1, 0, 0], [1, 1, 0, 0, -1, 0, 0, -1, -1, 1, 0, -1], [-1, 0, -1, 0, 0, 1, 1, 0, 0, 0, 0, 0], [-1, 2, 2, -1, 1, -1, 0, 0, 0, 0, 0, 2], [-1, -1, -2, 1, -1, 0, 1, -1, 1, -1, 0, -1], [-2, 1, 0, 1, 0, 1, 1, -1, 1, 0, -2, 0], [-2, 0, 2, -1, 3, -1, -1, 1, 2, 0, 0, 1], [-2, 1, -1, 1, 0, 0, 0, 0, -1, 0, 0, -1], [-3, -2, -1, 1, 0, 1, -1, 1, 0, 0, 2, -1], [-4, 1, 2, 1, 2, 0, -1, 1, 2, -1, 0, 3], [0, 1, -2, 0, 0, 1, 0, -1, 0, 1, 1, 0], [-1, 0, 1, -3, 1, -1, 0, 0, 0, 0, 0, 2], [-2, 0, 0, 0, 1, 0, 0, 3, 0, 2, 1, -2], [0, 0, -1, 0, -1, 0, 1, -4, -1, -1, -2, 0], [-2, 1, -1, 1, -2, 2, 1, 0, -2, 2, 0, -2], [-1, -1, -1, 0, -1, -2, -1, 0, -2, -1, 0, 0], [1, 0, 0, -1, -1, -1, 1, 0, 0, -1, -1, -2], [-2, -1, 1, 1, 0, -1, 1, 1, 0, -1, 0, 1], [-2, 0, 2, 1, 1, -3, -3, 2, 0, 0, 0, 3], [-1, 0, 2, 0, 0, 1, -1, 3, 0, 3, 1, 0], [3, -1, 1, -2, -3, 0, 2, -2, 2, -3, -1, 0], [-1, 0, -2, 1, -2, 0, 1, 0, -3, 0, 0, -1], [1, 1, 1, 0, -1, 0, 2, -1, 0, 0, -1, -1], [-3, 1, 2, -1, 0, -2, -3, 1, 0, 0, -1, 2], [-5, 1, -1, 2, -1, 2, 2, 0, -1, 0, -1, -2], [-3, 0, 0, -1, -2, 2, 1, -3, -2, 0, 0, -1], [-2, -1, 3, -3, 2, -2, -4, 6, 0, 2, 3, 0], [-6, -1, 0, -1, 0, 1, 0, 0, 0, 1, 0, -1], [1, -4, -1, 0, 3, 2, -1, 3, 1, 3, 3, -3], [0, 1, -1, -1, -2, 0, 0, -3, 2, -3, -3, 2], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, -2, 0, 2, 0, -2, 2, -2, 0, 2], [0, 1, -1, 2, -1, 0, -2, 0, -3, 2, 2, 2], [-8, 0, 1, 2, 0, 1, -3, -1, 2, -2, -1, 1], [-5, -1, 2, -1, 2, 0, 2, -2, 1, -1, -1, 1], [-4, 0, 1, 1, -1, -2, -1, -3, 1, -4, -3, 4], [-7, -2, 0, 3, 2, 1, 0, -1, 4, -2, 0, 0], [-3, 2, -1, 3, -2, 0, -2, -2, 0, 0, -1, 1], [-3, -3, -4, 1, 0, 2, -2, -1, 0, 0, 3, -2], [-1, -2, 1, 0, 2, 1, 0, 1, 0, 0, -2, 2], [0, -1, 0, 1, -1, -1, 1, -5, 2, -5, -3, 2], [-7, -1, 6, -1, 5, -4, -1, 6, 2, -2, 0, 3], [2, 0, 4, 1, 4, -1, 0, 5, 1, 3, 3, 0], [1, -2, 0, 2, 2, 1, 0, 3, -1, 2, 0, -3], [-2, 0, 2, -1, 3, -2, -1, 4, 2, 1, 0, 0], [4, -5, 1, -1, -2, -2, -1, 1, 1, -2, -1, 0], [-1, 0, -2, 0, 1, 3, 0, 1, 1, -1, 3, -2], [-5, -1, -1, -1, 0, -1, 2, 2, -3, 0, 3, -1], [0, -2, -3, 4, -1, 0, 2, -2, 5, -5, -4, 1], [0, 1, 2, -2, 1, 0, 1, 4, -2, 1, -1, -1], [-3, -4, -1, 4, -3, 1, 2, 2, -1, 0, 0, -4], [-3, 5, 0, -2, -3, 1, 6, -8, 0, -3, -4, 1], [-4, 1, -2, 0, 1, 3, 6, -5, 0, -2, -1, 0], [-3, -1, -4, 1, -2, 1, -1, -2, -2, -1, -2, 0], [6, 3, -2, -1, -3, 4, 4, -4, -6, 7, 0, -6], [-5, 1, -1, 3, 2, 0, 1, -6, 3, -3, -3, 2], [0, 0, 1, -2, -2, -2, 3, 1, -1, -4, -3, 1], [1, -3, -3, -1, -3, 2, -2, 0, -3, 4, 3, -2], [-4, 3, 0, -4, 2, -3, 1, 1, -2, -1, 3, 4], [-6, -1, -2, -2, -3, 0, 3, -4, 1, -5, 0, -1], [6, -4, -5, 0, 0, -1, -3, 3, 0, 0, 2, -3], [4, -6, -4, 2, -1, 2, 4, 5, 2, 0, 1, -6], [8, -4, 1, 2, -3, -3, 1, 3, -2, 0, 1, -1], [7, -4, 0, 2, 0, 2, -3, 0, 4, 4, 2, 1], [-4, -5, -2, -3, 2, -2, -5, 7, -1, 3, 4, -2], [4, 4, -4, 4, -5, 4, 6, -7, -1, -1, -5, -1], [0, 1, 0, 0, 4, -3, -3, 5, 3, 0, 2, 0], [3, 5, 0, -3, -4, 1, 4, -12, 2, -7, -8, 4], [4, -3, -2, 1, 3, -2, -4, 3, -3, 4, 2, -3], [-8, -6, 3, 1, 7, -2, -5, 5, 2, -3, -2, 5], [-5, -2, -5, 2, -4, 4, 2, 5, -4, 6, 8, -7], [2, 0, 0, -2, 1, -4, -1, 4, -2, 1, 5, 3], [5, -2, 0, -3, -4, 2, -2, -3, 1, 3, 4, -4], [-9, 0, 8, 1, 3, -3, -5, 2, 4, 2, 1, 5], [-11, -2, -1, -2, 1, 2, 0, 1, 3, 0, 0, 4], [-6, -2, 3, 0, 8, -4, -2, 7, 3, -1, 0, 5], [-5, 1, 0, 1, 5, 1, -3, 2, 0, 5, -1, 2], [12, -1, 0, -3, 1, 2, 1, -3, 1, -2, 4, 3], [-6, 1, 3, -1, 5, -3, -3, -1, 4, -4, -2, 4], [5, -1, -1, 3, -5, -4, 2, -6, -1, -5, -3, 2], [0, 0, -3, 5, -6, 6, 1, -5, -1, -3, -3, -6], [-3, 5, 2, -2, 0, 2, 0, 2, -7, 6, -1, -1], [2, 7, -4, 0, -7, 4, 8, -8, -3, 1, -1, -6], [2, 2, 4, 3, 0, -2, -3, 2, 1, 2, -2, -1], [-3, -3, 0, 3, -3, -3, 0, -6, -1, -9, -4, 6], [-9, 0, 4, -1, 4, 0, -1, 1, 0, -4, -1, 6], [8, 4, -2, 1, -3, 7, 1, -3, -2, 4, -2, -4], [10, 2, -6, -4, -6, 6, 5, -2, -8, 5, 5, -9], [-1, 0, -3, -2, -3, 2, 3, 0, -2, 4, 0, -6], [-6, 1, 0, -4, 4, -4, 0, 3, -2, -3, 2, 5], [-9, -4, 1, 7, 0, 0, 0, 0, 4, -8, -1, 5], [4, 0, -4, -1, -2, 5, 2, -7, 1, -3, -3, -3], [-7, 3, -2, -5, 0, 3, 2, -2, 0, 3, 4, 1], [8, -9, -5, 2, -2, 0, 1, 1, 3, -1, -2, -6], [4, 0, 0, 1, 2, 1, -3, 6, 2, 3, 6, -5], [-2, 3, -1, 5, 2, 2, -4, 3, 0, 7, -2, -2], [7, -4, 6, -4, 1, -6, -9, 7, 1, 2, 4, 1], [12, -4, 5, -4, 6, 1, -4, 9, 2, 7, 8, 1], [-4, 1, 5, 0, 3, -5, -4, 6, 7, 2, -2, 3], [-2, -2, 4, -3, 4, -4, -8, 4, -4, 2, 4, 5], [-8, 5, 3, 2, -2, -2, -2, -4, 0, -8, -6, 8], [2, 2, 3, -5, 6, -7, -2, -2, 1, -3, -2, 9], [1, 4, 8, 0, 4, 2, 1, -4, 5, 2, -1, 5], [-4, 5, 3, 3, 4, -3, -5, 6, -4, 6, 3, 5], [-5, 2, -2, 4, 2, -3, -4, 5, -6, 8, 4, -3], [4, 1, 10, 0, 5, -6, -2, 2, 3, -3, -9, 9], [0, -3, 2, -3, -1, -1, -2, 9, -4, 5, 3, -4], [-1, 2, 0, 3, 2, -2, -1, -3, 0, 2, 0, 0], [-2, 4, 3, -3, 0, -8, 1, -5, -2, -2, 0, 2], [0, 11, 4, 0, -1, 2, -2, 0, 1, 2, 2, 5], [12, -2, -8, 2, -12, -1, 4, -5, -11, 3, -3, -9], [-1, -3, -1, -2, 5, -2, 2, -2, 3, -1, -1, 0], [9, 7, -4, -1, -3, 0, 8, -10, 0, -2, -3, -1], [5, 7, -2, -4, -3, 1, 4, -4, -6, -2, -1, -4], [-3, 3, 1, 4, -4, -2, 0, 0, -3, 0, -5, -4], [-8, 3, -3, 4, -5, -2, 1, 0, -6, 3, -5, -5], [-6, 0, -4, -1, 2, 2, -5, 2, -1, 3, 4, -4], [3, -4, -8, 5, -2, 7, -1, -5, 3, -1, -1, -5], [10, -4, -6, -2, -8, 5, 4, -6, -2, -1, 1, -4], [4, 3, -3, -4, -2, -1, 8, -7, 0, -5, -5, 5], [12, 0, -1, 1, 3, 3, 3, 2, -1, 8, -2, -5], [2, 1, 5, -3, -3, -4, -6, 4, -4, 8, 4, 1], [6, -3, -2, -3, 6, -3, 1, 1, 2, -4, 6, -3], [12, -8, -4, 5, -6, -3, -1, 3, -2, 0, 0, -7], [-1, -2, 1, -2, 1, 2, -5, -4, 0, -1, -2, 3], [6, 0, -5, -3, -10, 4, 5, -4, -6, 3, 3, -6], [-7, 0, 2, 3, -4, 0, -5, 3, -1, 0, -1, -2], [0, 5, 8, -6, 3, -6, -2, 4, -5, 2, 3, 4], [1, 1, -4, -3, -1, 0, 0, -6, -7, 1, 3, -2], [4, -5, -3, 4, -7, 2, 1, -2, 7, 0, -2, -7], [-6, 6, 1, 1, -3, 1, -5, -2, -3, -2, 0, 2], [12, 1, -2, 6, -2, 1, 5, -2, -2, -2, -1, -5], [-4, -1, 6, 1, 4, -1, -4, -1, 7, -6, 0, 4], [2, 6, -1, -3, -1, 2, 4, 0, 1, 5, 3, -6], [-12, 1, 1, 6, 12, -1, -5, 7, 1, 2, 3, 2], [-9, -4, -8, -2, -5, -1, 4, -10, 2, -9, -7, 1], [1, -2, 0, 6, 1, -2, -6, 1, 5, 1, 1, 0], [-6, -8, -6, 7, -2, 7, -1, 2, 1, 0, -5, -8], [0, 2, 5, -8, 3, -7, -4, -6, 3, -7, -3, 11], [-1, 2, 7, -3, 4, -3, 1, 4, -6, 3, -1, 1], [9, 0, -6, -3, -6, 11, 5, 1, -3, 6, 5, -6], [3, 2, -4, 3, -2, 5, 13, -9, 2, 2, -6, -2], [10, -3, -1, 8, -4, -1, -5, -1, 3, -5, -4, 3], [-1, -5, 1, 4, 1, -3, 1, 2, 8, -8, -8, 0], [5, -8, -4, -1, 2, 1, -2, -10, 6, -7, 2, -2], [0, 4, 2, 3, -5, -2, 2, -7, -5, 0, -3, -3], [-8, 2, -2, -1, 0, -1, 3, -4, 1, -8, -11, 6], [5, -2, -1, -7, -1, 4, -4, 10, 1, 6, 10, -6], [-13, 2, 6, 0, 3, -4, -2, -1, -5, -3, -3, 8], [5, 1, -4, -2, 0, 1, 6, 0, -9, 5, -4, -6], [12, -6, -3, -1, 2, 2, 5, 5, -1, 6, 2, -7], [-3, -4, -1, -1, 2, -2, -2, 4, 7, -4, 2, -2], [2, -1, -3, 5, 0, 6, 2, -5, 3, 3, -2, -1], [7, 3, -6, 3, -3, 3, 2, 1, -1, 5, 5, -8], [0, 2, 3, -8, 6, -2, -1, -5, 2, 0, -3, 5], [-9, -11, 1, 5, 0, -1, 1, 0, 7, -7, -1, 1], [-7, 0, 3, -2, 4, 0, -3, -1, 3, 0, -4, 9], [5, 10, 4, -3, 1, 1, -2, -4, -3, 6, -1, 8], [14, 7, 0, -9, -6, 2, 3, 0, -8, 10, 10, -11], [-4, 6, 3, -9, 6, 0, -4, 7, 0, 4, 7, -1], [5, 1, -2, -1, -6, -7, 2, 1, -4, 1, 1, -5], [-9, -5, 4, 6, 11, 0, -5, 1, 14, -6, 0, 9], [-6, -5, 0, 2, -3, 10, 6, 4, 0, 4, -1, -4], [2, -1, -3, 5, 6, 2, -3, 2, 3, 0, -2, 0], [8, 2, 1, -3, -5, 3, -3, -5, -9, -3, 0, 2], [7, 3, -4, -1, 0, -1, -2, -5, -4, -5, 1, 3], [2, 0, 4, -4, 3, -12, -8, -1, 2, -5, 2, 3], [7, 3, 1, 1, 2, 3, 5, -4, 3, 4, 2, -1], [-13, -2, -5, -3, 0, 5, 7, -10, 5, -9, -3, 0], [10, -2, 3, 4, 3, 1, -3, 10, -1, 7, 5, 4], [-4, 2, 4, 1, 4, -3, 1, -3, -1, -6, 1, -1], [4, 5, -7, -3, 3, 4, -1, 2, -8, 7, 8, -9], [-4, 0, 5, -1, 4, 2, 4, -3, 0, -6, -2, 4], [1, 4, 7, -14, 4, -9, -1, 0, 2, -5, -4, 9], [-4, -2, 9, -3, 9, -4, -3, 0, 5, -4, 3, 10], [21, 10, 0, 4, -9, 6, 1, -1, -7, 5, -1, -10], [-13, -1, -4, 5, 0, 9, 2, -5, 4, -3, -10, 4], [-5, -3, 2, 4, 4, 0, -3, 3, -2, -7, 2, 4], [-11, -1, 14, -2, 7, -9, -4, 8, 3, -3, -2, 9], [12, 14, 3, -4, -8, 4, 3, -7, -11, 1, -5, 3], 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4, -12, 1, -7, 2, 4], [5, 0, -17, 5, -22, 4, 1, -6, -9, 7, -3, -10], [-13, -7, -3, 1, -1, 11, 4, 9, -2, 3, 2, -7], [4, 11, 7, 7, 3, -8, -1, 0, -6, -2, -4, 0], [24, 1, -10, -8, -9, 3, 4, 0, -13, 11, 10, -2], [-19, -5, -6, 7, -11, 0, 7, -2, 0, -11, -4, 0], [-11, -14, -2, 2, 11, 0, -16, 13, 10, 1, 11, -1], [37, 5, 0, -9, -5, 2, -3, -4, 2, 13, 11, -2], [9, -9, -2, 0, 7, 7, 7, -3, 4, -16, 2, 1], [-27, -18, -4, 0, 2, 2, -8, 1, 18, -21, 0, 6], [-1, -1, -3, 7, 9, 9, -7, 13, 2, 11, -1, -1], [-2, 9, 4, 0, 2, -5, -1, -8, 2, -7, 3, 2], [-10, 5, 13, -10, 12, -18, -4, 11, 11, 0, 4, 8], [18, 10, 7, -1, -8, 11, 9, -4, 2, 16, 0, -7], [13, 10, -2, -7, 3, 3, 2, 1, -9, 18, 12, -7], [6, 20, 9, -6, -1, -10, -7, 1, -11, 10, 1, 6], [-9, 10, -6, -3, 3, 5, 7, -9, 6, 1, -7, 3], [-17, 10, 3, 4, 7, 3, 2, 4, -3, 1, -7, 10], [-12, 6, 2, 2, 3, 8, 3, -1, -12, 8, 1, 3], [17, 13, 8, -7, -1, -2, 1, -24, 5, -9, -19, 19], [-23, -6, 3, 1, 7, 5, 0, 6, 19, -12, 1, 12], [-2, -13, 6, 8, 9, -8, -3, 9, 11, -10, -13, 2], [2, 4, 1, -7, 1, -9, -4, -6, 2, 4, 1, 2], [14, -9, -17, 11, -13, 7, 12, -22, -3, -10, -9, -13], [-3, -3, -4, -2, -3, 13, 9, -2, -12, 10, 11, -10], [23, 15, -14, 10, -15, 10, 12, -18, -21, 10, -1, -17], [7, -7, -10, 3, -10, 10, 10, -6, 0, 6, -9, -15], [-20, 5, 3, 0, 10, -6, -8, 9, 4, -2, -1, 14], [-16, -6, -8, -5, -4, 1, -4, -2, 3, -11, 9, 5], [14, 14, -7, -8, -11, 15, 14, -10, -16, 18, 9, -12], [8, 2, -1, 6, 7, -3, -3, 10, 2, -2, -1, 6], [14, -13, 2, -5, 4, -7, 1, 14, -4, 12, 4, -4], [0, 7, -11, 3, 3, 5, -2, -13, 6, -4, -5, 2], [-3, 19, -1, 4, -4, -1, -7, -6, -3, 2, -7, 6], [-9, 1, 1, -5, -4, -7, 2, -10, 0, 0, -6, -2], [-25, 2, 10, 1, 11, -8, 2, 4, 14, -1, 1, 11], [16, -4, 1, 0, 1, -8, -1, -2, 8, -17, -1, 12], [-5, 6, -1, 7, -5, 12, 5, 7, -8, 9, -1, -9], [-15, 9, -8, 13, -2, 6, 0, 5, -7, 20, 15, -10], [-10, -1, 5, 11, 2, -2, 4, -13, 6, -9, -6, 6], [-20, 2, -3, -8, -5, -6, 1, -9, -8, -12, 2, 10], [10, 2, -8, 1, -14, 1, 0, -1, -9, 7, -4, -5], [-12, 3, 5, -13, 16, 0, -9, 25, -9, 16, 19, 6], [-2, 21, 10, -6, 7, -16, 0, -4, -5, -4, 5, 3], [17, -17, 0, 2, -6, 1, 8, 16, -9, 10, 4, -17], [-5, -5, 10, 2, 15, -14, -17, 23, 5, -4, 8, 13], [-14, 9, -5, 3, -14, 19, 14, 3, -6, 8, 0, -12], [15, -12, -2, 3, -1, -8, -13, 7, 1, -6, 2, -3], [-4, -14, 6, 1, -10, -6, 11, 4, -3, -5, -1, -13], [22, 17, 1, -6, -1, 2, 17, -15, 4, 3, -13, -2], [-3, 11, -3, 9, -13, -9, -5, -1, -8, 3, -9, -2], [16, -11, -15, -4, -22, 3, 12, -12, -6, -7, -3, -11], [-3, -18, -4, 3, -11, 1, 3, 3, 1, -11, 1, -10], [4, 9, 10, 5, 1, 0, -6, 6, 0, 4, 12, 4], [1, -10, -12, 9, -14, 7, 4, 5, -7, 6, -2, -14], [-10, 3, -6, 9, -2, 1, 2, -1, 6, -4, -5, 0], [-8, 2, -1, 7, -3, 1, 0, -10, 5, -18, -11, 14], [3, 0, 0, 0, 0, -6, -5, 9, -4, 5, 6, 11], [-3, 15, -4, 3, 5, 5, 8, 3, -11, 6, 0, -5], [-40, 12, 6, -5, 7, -1, 2, -4, -7, 5, 7, 0], [-9, -11, -1, 5, 15, -16, -12, 9, 12, -7, 2, 4], [-19, 11, -2, 12, 0, 5, -2, -4, -9, 6, 4, 1], [14, -2, 8, -11, 12, 7, 5, 4, -1, 9, 2, -2], [6, -5, 18, -8, 10, -11, -14, 9, 2, -6, -3, 13], [-22, -1, 20, -4, 20, -14, -9, 18, 10, -8, 5, 21], [-1, -15, -7, 1, -13, 1, 8, -18, 7, -22, -11, -2], [-3, -4, -12, -1, -7, 5, 5, 6, -2, -1, 8, 4], [-11, -7, 0, -6, 7, -1, 8, 1, 10, -10, -12, 6], [1, 5, 6, -10, 2, 4, 4, -4, -1, 12, 1, -7], [-13, 9, -6, -12, -4, 5, -6, 14, -11, 13, 16, -3], [-10, 7, -1, -5, 0, -6, -5, -6, -4, -14, -9, 8], [-2, -2, 11, 0, -1, -4, 4, -6, 3, -1, -7, 18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8046_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8046_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_8046_2_a_n(:prec:=12)
    chi := MakeCharacter_8046_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_8046_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_8046_2_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8046_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-3, 594, -1607, -345, 3573, -1551, -1561, 684, 334, -82, -31, 3, 1]>,<11,R![142707, 697652, 552829, -260965, -297080, -1779, 46139, 7069, -2421, -638, 11, 14, 1]>],Snew);
    return Vf;
end function;