
// Make newform 8046.2.a.m 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_m();".
// 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_m();".
// 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 := [-375, 910, 761, -2249, -417, 2013, -63, -774, 106, 116, -21, -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], [1592878652, -3729321842, -2256579470, 6463643167, 447932378, -3495027303, 410288818, 609086107, -109537587, -27408721, 5673372, -32197], [-4301878920, 9660687071, 6698494966, -16685502287, -2097844062, 8970420017, -664646379, -1586491494, 222792211, 77892894, -12051765, -337779], [1309434950, -5435421687, 2382541943, 8040696879, -6124911496, -3268300794, 3479105328, 137663433, -610246892, 68317507, 27982820, -4626967], [2800717475, -8170221548, 594551152, 13075477756, -6551037399, -6005098721, 4419720832, 526825367, -818338273, 69519048, 38266875, -5834388], [-6906022130, 12966000711, 17416329301, -22956311877, -15262968602, 13696849342, 5470397091, -3330885579, -795142464, 329349654, 33647510, -11071809], [-1289750025, -1591143977, 13154737748, 804575664, -19399390581, 1916697621, 9679352383, -1697104197, -1640249962, 327537312, 74241930, -15600107], [-6112777815, 18959758726, -3005054784, -30654436132, 17633361118, 14014610012, -11652307114, -1047579084, 2153825106, -207068641, -100688050, 15915726], [-38973036, -842422122, 2366997820, 1171309316, -3967835268, -179845605, 2107854700, -254489192, -366093260, 64707528, 16732001, -3317302], [5205009495, -6697290927, -19695978447, 13297311074, 22410652829, -9615020874, -9796137762, 3159684708, 1573561298, -418005468, -69687500, 17078923], [-395658693, 2627858133, -3278532055, -3951467976, 6428622786, 1341737333, -3586733161, 238661630, 632612308, -98087514, -29080114, 5445786]];
    Rf_basisdens := [1, 1, 5855999, 29279995, 29279995, 29279995, 29279995, 29279995, 29279995, 5855999, 29279995, 5855999];
    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], [0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1], [-1, 0, 0, 0, -1, 0, 1, 0, -1, -1, -1, 0], [-1, 0, 2, -1, 0, 0, 1, -1, 0, -1, 0, -1], [-2, 0, 0, 1, 0, -1, -1, 0, -1, 1, 0, -1], [-2, 1, 0, 1, 0, 0, -1, 0, 0, 1, 1, 0], [-3, -2, 0, 0, -2, 0, -1, 0, 0, 0, -1, -1], [-1, 2, 0, 0, 1, 2, -1, 1, 1, -1, 0, 0], [-2, 1, 0, 0, 0, 1, -1, 1, -1, 0, 0, 1], [-2, 1, 2, -1, 0, 1, -1, 0, 1, 1, 0, 0], [-1, 1, 1, 0, 1, 1, 1, 0, 0, 1, -1, 1], [0, 1, 0, 0, 2, 1, -1, 1, 2, 1, 1, 1], [-2, 0, 0, -1, -1, 0, -1, 2, -1, -1, 0, -2], [0, 0, 1, 0, 2, -2, 3, -2, -1, -1, 0, 2], [-2, 1, -2, 1, -3, 0, -1, 1, 1, -1, 1, 0], [-3, 0, 0, 1, 0, -1, 0, 0, -1, 0, -2, -1], [-3, 1, 1, 1, 2, 1, -1, -2, 1, 1, -1, 2], [-1, 0, 1, 0, -1, 1, 2, 1, 1, -1, 1, -1], [0, 0, -1, -1, 0, 1, 1, -2, -1, -1, 0, 0], [0, 0, -1, 0, 1, -2, 0, 0, 1, 1, 0, 1], [-2, -3, 2, -2, -1, -1, 4, 0, -1, 0, 0, -3], [-1, 1, 1, 0, 1, 1, 2, -1, 1, 1, 0, 2], [-6, -1, -2, 2, 0, 0, 2, 2, -2, -1, 1, 0], [1, 0, -2, -1, 1, 0, -1, -1, 2, 0, 1, 1], [-4, -3, -1, 2, -3, -2, 3, -1, -2, -3, 0, -1], [2, -2, 1, -2, 0, -2, 0, 0, -1, -1, 0, -2], [-1, 0, 2, 0, 1, -1, -1, -2, 0, -1, -1, 1], [1, -2, -2, 0, -1, 1, 1, -3, -1, -2, -1, 1], [0, -1, -4, 1, -2, -4, 1, -2, -4, 0, -1, 2], [-3, -3, -2, 1, -3, -5, 0, -1, -1, -1, -2, -1], [-1, 2, -2, 0, 1, -1, -4, 2, 1, 2, 1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, -4, 0, -3, -3, 0, 2, 0, -4, 2, -1], [0, 3, -5, 1, 0, 0, -2, 2, 2, 0, 4, 2], [-2, -2, 0, 1, 0, 1, 0, 3, 2, -1, 3, -2], [3, -2, -1, -1, 1, 0, -1, -1, -2, 5, 2, 0], [1, -6, 2, -3, 1, -4, 3, -5, -2, 1, -2, -1], [-3, -3, 1, 0, 0, -3, 4, 0, 1, 1, 0, -1], [8, 0, 3, -1, 3, 4, 3, -2, 2, -1, -2, 3], [-7, 2, -1, 0, 0, 4, 2, 4, -1, -2, 1, 1], [3, 1, -1, -1, 1, -2, 1, -3, -1, 0, -3, 2], [-9, 1, -2, 2, -1, 5, -3, 4, 1, -1, 1, 2], [2, 2, -1, 1, 1, 1, -6, 4, 2, 5, 3, 1], [4, -1, -1, -1, 1, 2, 2, -1, 2, 0, 2, 3], [-3, 1, -3, 2, 1, -3, -5, 3, 2, 3, 4, -1], [-2, -2, -1, -2, -1, 4, 0, -1, 1, -4, -2, 1], [-7, 2, -2, 4, -3, 6, -3, 4, -2, -3, -4, 2], [-7, -4, 3, -1, 1, 2, 5, -3, -2, -3, -1, 0], [-1, -3, -2, 0, -6, -2, 1, -1, -1, 1, -1, -5], [3, 3, 1, 0, 2, 5, -2, 3, 3, -3, -1, 4], [-7, 5, -2, -1, 1, 3, -5, 7, 3, 3, 2, 1], [2, 9, 1, 2, 6, 1, -2, 0, 1, 4, 1, 2], [1, 0, -2, 1, 2, -2, 2, 0, 4, 1, 4, -1], [-1, -4, -5, 1, -2, -3, 3, 0, -4, -5, -5, 2], [4, 1, 1, -2, 1, 7, -2, 2, 2, -4, -3, 0], [-1, 2, 5, 0, 4, 2, 1, 0, 0, 7, 3, 1], [-4, 0, 0, -2, -1, -4, -2, -1, 0, -1, 0, -3], [-1, -2, -1, 0, 1, -1, -2, -2, -1, 5, 4, 0], [2, -8, -5, 1, -4, -4, 1, -1, -3, -4, -1, -4], [1, -8, -2, 0, -9, -5, 2, 0, -3, -4, -2, -3], [2, 1, -3, 2, 0, 2, -3, 0, 4, -2, -2, 3], [1, -5, -4, 0, -2, -1, -2, 1, 1, -6, -3, 0], [-6, 0, -3, 4, -2, -2, -5, 2, -1, -1, -4, 0], [-2, 1, 4, 0, 2, -1, 0, 0, 0, -1, -1, -3], [-2, 2, 3, -1, -1, 1, -7, 6, 4, 2, 0, -4], [5, -1, -3, -1, 0, 1, -3, 2, -1, 4, -2, -1], [-1, 4, -2, 3, -1, 2, -1, 4, 2, 0, 3, 0], [-5, -2, 4, -1, 2, 0, 0, -3, -4, 5, 2, -3], [-1, 2, 1, -2, 4, -3, 1, -7, -1, 1, -4, 4], [-5, 0, 1, 1, 3, -2, 2, -3, -3, 0, 0, 2], [3, -3, 5, -3, -2, 0, 3, -3, 0, -3, -5, -1], [6, -3, 0, 3, 1, -2, 1, 0, 1, 3, -1, 2], [-3, -2, 2, 3, -1, -5, 3, -7, -3, -1, -4, 1], [-7, -1, 5, -1, 1, 3, -2, 3, 0, -3, -3, -5], [2, -3, -3, 3, -7, 2, 0, 2, 2, -2, 0, 1], [0, 4, -2, 3, 0, 4, 3, -3, -4, -4, -2, 7], [2, -4, 2, 1, -4, -1, 1, -5, -3, 1, -1, -4], [-7, 3, 1, 2, -1, -1, 2, 3, -1, 1, 1, -1], [-4, -1, 5, -3, 2, 0, 1, -2, -4, 3, 3, -2], [-5, 2, 3, 1, 4, -2, 1, 4, 2, 4, 7, -3], [7, -4, -2, 3, 2, 0, 1, -5, -1, 0, -1, 0], [2, 3, 3, -2, 7, 0, -1, -2, 2, 1, 2, 0], [2, 2, 6, 0, 7, 0, 2, -4, 4, 4, 5, 0], [-6, 2, 0, 0, 3, -1, 2, 2, -3, -1, 2, -1], [3, 1, 4, 0, 4, -2, -2, 0, 4, 1, 3, -2], [1, 0, -4, -1, 1, 0, -1, 3, 1, 5, 4, 1], [-4, 2, -1, 0, 1, 6, 1, 7, 2, -3, 2, 3], [-3, 9, 5, -3, 5, 1, -3, 0, 4, 2, -1, 2], [-5, -5, -2, 2, -5, -5, 1, 0, -7, 3, 0, -4], [6, -4, 0, -3, 0, -7, 3, 2, -3, 1, -1, -6], [2, -4, -1, -3, -2, 3, 1, -4, -1, -1, -3, 1], [-2, -2, -1, 3, -1, -4, 1, -2, -6, -3, -4, -3], [-4, 2, 8, 0, 5, -3, -2, 0, 3, 5, -1, -1], [1, -1, 4, -1, -2, 6, -1, 2, 2, 2, -5, 0], [-1, 9, 5, -4, 7, 6, 0, -1, 5, 3, 3, 2], [1, -9, 2, -3, -5, -4, 0, -1, -1, 5, -2, -6], [8, -5, 2, -3, 1, -3, 4, -6, -4, 1, -3, 0], [-7, 2, -1, 2, 0, 2, 0, 1, -2, 2, 2, 4], [-7, 0, -5, 4, -5, 1, -3, 0, 2, -1, 2, 2], [1, 5, 4, -3, 4, 4, -4, 1, 0, 1, -6, 4], [-14, -1, 5, 0, -4, -5, 5, -2, -7, -4, -6, -8], [8, -7, 0, 0, -3, -1, 8, -4, -6, -4, -4, 3], [4, 3, -3, 1, 0, -4, 3, -3, -4, 2, 5, 5], [9, 2, 0, 0, 6, -4, -1, -3, 5, 5, 4, 6], [-5, 5, 3, -2, -6, -1, -6, 1, 4, 0, 0, -5], [-5, -6, -4, 0, -10, 2, -1, 0, -4, -7, 2, -9], [10, -1, 3, -5, -1, -2, 0, -1, 0, 0, -1, 1], [0, 2, -4, -2, -4, 9, 0, 9, 2, -6, -2, 2], [7, 3, 3, 1, 7, 2, 3, -3, 1, 0, -2, 3], [6, -7, -3, -3, -10, -5, -1, 3, -7, -4, -4, -6], [2, -2, 2, -6, -1, 2, -4, -1, 1, -1, -1, 1], [0, 6, -8, 5, -3, 5, -5, 3, 1, -6, 0, 5], [-3, 2, 0, 0, 6, 0, 0, -4, 5, 3, -4, 6], [0, 2, -3, 2, 2, 5, -1, 5, 0, 1, 2, 1], [5, -6, 8, -7, -2, -5, 6, -3, -2, -3, -3, -3], [2, 3, 7, -3, 6, 2, -6, -1, 3, 11, 2, -1], [-7, 1, 0, 1, -6, 4, -3, 5, -1, 1, -4, -6], [8, 2, 4, 1, -2, 5, 2, -3, -5, -2, 2, -2], [8, -5, 0, -2, 0, -5, 3, 1, -2, 4, 2, -3], [3, 4, 3, 0, 4, -1, 6, -4, 2, 9, 3, 4], [-6, 6, 3, -1, 7, 5, -1, 2, 7, 2, 2, 2], [-14, -1, 0, 0, -5, 0, 4, -3, -3, -3, -4, 2], [-14, -1, -3, 3, -9, 0, -7, -1, -3, -2, -4, -4], [-5, -2, -2, -1, -9, 4, -1, 2, 1, -5, 0, -1], [-13, -1, 6, -5, -7, -2, -3, 4, -1, 1, 0, -5], [-3, 7, -2, 2, -2, 0, -5, 9, 4, 2, 4, 2], [3, 2, -3, 4, -4, -3, 0, 4, -4, -3, 0, -3], [8, 5, -2, 2, 0, -1, -5, -2, 4, -1, 0, 5], [-3, 0, 3, 0, -1, 1, 6, 0, 3, -1, 6, -3], [19, -6, -2, -3, 3, -7, 7, -9, 3, 2, 1, 1], [9, 1, -2, 2, 5, 3, -5, 0, 0, 10, 4, 4], [6, 2, -3, 5, 7, -3, 4, -2, -2, 2, 1, 6], [5, 7, 7, 3, 5, 4, 1, 0, 4, 7, 7, 2], [4, -7, 2, -2, 1, -6, 2, 1, 5, 5, 4, -1], [2, 7, -2, 5, 4, 6, -5, -5, 7, 2, 3, 10], [6, -3, -2, 0, -1, -3, -8, 5, 1, 6, 2, 0], [-4, 4, -6, 4, 2, 0, -6, 2, -3, 6, 6, 3], [4, 3, -3, 2, -7, -1, -5, 7, -1, 3, 4, -10], [-3, 0, -7, -3, -2, 0, -2, 3, 7, 3, 7, 0], [3, 5, -1, 0, 2, 6, -4, -1, 7, -2, 2, 8], [-3, 2, 0, 2, 6, -3, 4, -1, -4, 0, 0, 3], [9, 4, 4, -1, 4, -2, 3, -4, -1, 5, 3, -3], [2, 0, 1, -6, -3, -1, -1, -1, 0, -2, -5, 1], [-10, -3, -4, -2, -7, 4, -6, 4, -1, -4, -1, -10], [-13, 2, 6, -2, -6, 4, -1, 6, 2, -1, -2, -4], [5, 3, -5, 6, 8, -3, 3, 3, -1, 7, 6, 6], [4, -2, 1, -6, 1, 4, -8, 3, 2, 3, 2, 0], [3, -4, 3, -1, -2, -2, -8, 1, -3, 1, -3, -12], [-15, 2, 1, 4, -2, 2, -7, -1, 0, 5, 1, -8], [14, 9, 1, 4, 9, 5, -1, 3, 5, -3, 0, 8], [5, 5, -2, 7, 1, 0, -4, 1, 5, 6, 5, -2], [-14, -3, 7, -2, 1, -7, 5, -5, 2, 1, -3, -2], [3, -1, 7, -7, 4, -4, 3, -4, 2, 3, 1, -7], [12, -11, -8, -4, -4, 3, 2, 4, -5, 0, -4, 0], [9, -5, 5, -8, 1, -6, 10, -4, -2, -2, 0, -6], [21, -1, -4, 2, 0, -2, 4, 1, -3, -1, -2, 6], [11, -8, -1, -4, 3, -8, 12, -10, -2, 4, 6, -1], [3, -3, -5, 1, -7, -3, 2, 0, -2, 2, 6, -4], [-3, 0, -13, 6, 3, -4, -3, 3, 0, 0, 2, 1], [3, -3, 7, -4, 9, 5, 4, -5, -2, 7, -1, 4], [10, -2, 8, -3, 3, 4, 12, -5, 0, -7, -6, 2], [9, -4, 5, -4, 0, 6, -2, -3, 7, -3, 1, -1], [7, 4, -8, 0, -5, 7, -6, 4, 2, -14, -4, 2], [1, 9, 2, -4, 0, 8, -6, 7, 3, -3, -2, 3], [-3, -3, 11, -3, -1, -4, 4, -3, 3, -4, -2, -3], [10, 2, -3, 5, 1, -5, -1, -5, 0, 0, 1, 5], [9, 10, -5, 2, 12, 2, -2, 1, 7, 11, 5, 9], [4, 2, -7, 2, 6, -1, -7, 1, 3, 5, 9, -2], [-3, 1, -2, 3, -5, -6, -4, -2, -7, 5, 2, -7], [-19, 6, 4, 5, 1, 3, 0, 1, -9, 1, -1, -3], [-10, 5, 0, 1, 3, 9, -9, 5, 5, 2, -4, -4], [10, -9, 9, -8, 2, -3, 1, -5, 12, 0, 2, -5], [-6, 10, -8, 11, -2, -4, -10, 4, 0, 3, 7, 0], [-9, -2, 3, 1, -9, 1, -2, -4, 0, 3, 1, -6], [6, -1, -2, -2, 0, 1, 11, -2, 3, -4, -3, 5], [0, -6, -7, 3, -5, 6, 4, 2, -2, -9, -2, 0], [-7, 2, -5, 2, -4, -1, 2, 4, -4, -4, -5, 2], [-16, 6, 0, 0, 2, 7, -8, 6, 1, 3, 0, -5], [-2, 12, 2, 2, 9, 0, -6, -2, 9, 11, 11, 3], [9, -5, 5, -3, -3, -7, 12, -7, -4, -3, 4, -1], [-2, 14, 0, 6, 1, 11, 4, 2, -1, -7, 0, 8], [3, -3, 9, -6, 0, 1, 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-2, 3, -8, -10, 3, 2, 3, 8, -3], [-4, 0, 11, 0, 5, -3, -2, -6, 10, 8, 7, 3], [-33, 15, 0, 4, 6, 11, -6, 15, 1, -10, -2, 4], [22, 0, -10, 8, 11, 14, -8, 6, 5, -2, -3, 17], [-13, 3, 1, -2, 7, 16, 0, 13, 7, -6, 4, 11], [-42, 5, 5, -3, -10, -9, -1, 1, -8, -8, -1, -14], [0, -10, -13, 3, 1, -4, 4, -13, -2, -13, -4, 5], [-10, -9, 10, -7, -5, -2, 2, 4, 3, 4, -5, -12], [-15, -10, 4, -2, -10, -2, 3, -1, -14, -10, -2, -23], [18, 14, -5, 5, 14, 1, -3, -7, -2, 10, -2, 5], [-20, 2, -7, 8, -5, 1, -9, 11, -1, -6, 4, -6], [-1, 20, -8, 2, 14, -4, -11, 2, 4, 6, 4, 4], [6, -3, -5, -1, 4, 9, -2, 9, 7, 1, 10, -3], [-11, 6, 5, -2, 5, 5, -12, -5, 12, 6, 1, 2], [-5, 20, -8, 6, 15, 4, -10, 16, 8, 14, 7, 11], [4, -1, -1, -5, 5, 8, 1, -1, 10, 2, 16, 4], [-10, -14, 0, 1, -10, -7, 8, -6, -10, -5, -15, -6], [-5, -2, -16, 11, 3, -7, 2, -12, -3, 9, -2, 18], [16, 2, 3, -10, 8, 2, -2, -9, 13, -4, 7, -1], [21, 22, -1, 2, 8, -7, -15, -1, 12, 14, 12, -2], [3, 3, 12, 6, 0, 3, 8, -8, -12, -4, -7, 0], [12, -9, 8, -12, -3, 0, -1, -4, 0, 0, 0, -8], [-16, 14, 8, 3, 0, 4, 3, 0, 13, -5, 3, 2], [-24, 9, 2, 3, -1, -7, 13, -6, -3, -6, -2, 3], [2, 9, 4, 1, 7, -18, 0, -5, 0, 12, 3, 3], [2, -1, -5, 5, 8, -7, -5, -12, 4, 17, 4, 0], [-26, 8, 7, 7, -1, -1, -1, 5, 5, -9, 2, -10], [3, 10, -3, 6, 1, -13, -15, 5, 15, 9, 11, -2], [20, -17, 12, -9, -3, 0, 6, -3, -4, -1, 1, -4], [-15, -8, 8, -3, -8, 6, 9, 0, -5, 4, 2, -8], [4, 5, -20, 10, -1, 9, 0, 12, -4, -4, 1, 10], [-3, -2, 3, 2, 1, 0, 8, 5, 2, -10, -6, 3], [-20, 1, -17, 16, -9, 3, -5, 6, -9, 4, -4, 1], [-12, -10, 8, 0, -15, 4, 17, -9, -11, -21, -18, -3], [-5, 5, 0, -8, -8, -1, 9, -3, -2, -13, -4, -11], [-19, -17, -3, -6, -16, -9, -1, -2, -8, -2, 5, -14], [17, 14, 3, 6, 14, 16, -9, 7, 19, 10, 3, 17], [5, 7, 4, 4, 11, -1, 6, -17, 5, 2, -7, 13], [-2, 15, -4, -1, 5, -15, 4, -10, 0, 3, -3, 0], [0, -12, -1, 1, -10, 3, 2, 11, 7, -5, 1, 1], [-7, 3, 16, -7, 13, 13, 10, -2, 2, 6, 1, 7], [-23, 6, 0, 2, 7, 8, 1, -6, -6, 3, -8, 4], [1, 5, 8, -1, 9, 10, 0, 17, 10, -4, 9, -1], [-28, -4, 0, 0, -11, -7, 0, 0, -2, -3, -2, -16], [27, 11, -3, 4, 6, -4, -4, 3, 10, -4, 4, -4], [8, -8, -5, -2, -7, -8, 0, -8, 1, 15, 8, -8], [-17, 8, -5, 11, 18, -4, -3, -8, 8, 13, 11, 14], [-15, 4, 11, -5, 12, -6, 10, -4, 5, -10, 9, 6], [-36, -2, -2, -2, 2, 7, -6, 1, 2, -8, -7, 1], [-16, 1, 12, -8, 12, -6, 3, -14, 3, 17, 3, 1], [5, 8, -4, 3, 15, 1, -16, -1, 1, 28, 16, -3], [14, -3, -7, 2, -14, 2, 1, -7, -11, -5, -13, 6], [-20, 6, 14, -4, 6, 0, 5, -16, 0, -5, 1, -7], [-21, 4, -2, -6, -17, 16, 0, 11, -1, -19, -12, 1], [5, 15, -9, 3, 2, -9, 2, 4, 4, 6, 4, 11], [22, 16, 15, 0, 22, -1, -7, 5, 22, 14, 20, 4], [4, 20, -3, 7, 7, 1, -14, -3, 4, 6, 0, 1], [-1, 1, 11, -2, 3, -3, -4, 1, 10, 14, 15, 8], [-21, -2, -6, 8, -5, 4, -1, -1, -16, -4, -6, -5], [7, 6, 10, -6, 9, -1, 6, -3, 4, 7, 11, 1], [-6, 2, 8, -2, 3, -1, 8, -2, -6, 7, -8, -2], [-27, -1, 10, -10, 8, 4, 14, 1, 0, -8, 3, -8], [-37, 12, 2, 7, -12, 10, 1, 8, -11, -12, -12, 1], [-6, 16, 1, 0, 7, 5, 15, -12, -1, -15, -2, 14], [-9, -1, -3, 8, -8, -7, 7, -3, 2, -14, 6, -10], [8, -15, 25, -9, -13, -1, 14, -15, -6, 0, -17, -7], [-7, -7, 1, -2, -14, -7, 7, 0, -9, -10, -7, -16], [-14, 12, 18, 0, 1, 10, 8, -8, -10, -12, -7, 0], [3, -6, -11, 0, -9, -1, -4, 4, 2, -9, -8, -6], [26, -9, -6, -4, 2, -3, 10, -2, 2, -7, 4, 5], [8, -1, 2, -8, 8, -12, -2, -10, -1, 15, 0, -2]];
    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_m();".
// 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_m(: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_m();".
// 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_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8046_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-375, -910, 761, 2249, -417, -2013, -63, 774, 106, -116, -21, 5, 1]>,<11,R![101487, -2640, -374243, -453995, -184540, 12387, 30775, 6231, -1093, -486, -17, 10, 1]>],Snew);
    return Vf;
end function;