
// Make newform 8046.2.a.q in Magma, downloaded from the LMFDB on 24 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_q();".
// 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_q();".
// 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 := [-7083, 6282, 53551, 9808, -55580, -12990, 24455, 5066, -5559, -882, 680, 70, -42, -2, 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], [19597736159910, -36206262335215, -143402427852725, 23331243759189, 149813081971647, -11816541353639, -63433606982404, 6907753319984, 12477302034149, -1870674412587, -1106373672733, 202430140259, 33914318185, -6885747582], [34191931183770, -56298435608683, -232233141936539, 63297600417105, 234011215846473, -40945399675328, -95500280471449, 16571607563132, 18377007078962, -3518878903680, -1615645352599, 340112131925, 49436090470, -11004140892], [53337950263866, -68967721612466, -401075285306924, 69177996434839, 418064433687984, -54385976946928, -173410067107969, 26870136857865, 33227082074766, -6118651043878, -2872804185614, 595251271041, 86554061461, -19118330939], [49744016665776, -56296105207467, -377615249540662, 35993911720081, 426625795784475, -34950317548215, -191223255135815, 24638483686895, 38941941028796, -6629286791539, -3523804015461, 703785403742, 109271241761, -23639242991], [105376014149448, -135681262744841, -696858166252685, 189622681806508, 659898648295785, -135245303862964, -250882871913973, 50850635857116, 45107127742866, -9624887605363, -3739110618812, 842563500174, 109910586859, -25609954388], [-101151393317133, 132289761077346, 688027259578091, -192408481665842, -694429136478564, 143968104585861, 279816457466485, -57437914514824, -52596458855725, 11379461904392, 4499074114704, -1026991737247, -134861372299, 31764401845], [-113370111470034, 162909865954943, 817249872781809, -203381316447602, -827159607477825, 148254476159677, 334462797242739, -61998638358803, -63127448399537, 12830895743357, 5420978081009, -1191384212828, -162933719250, 37435189105], [-109196500860438, 153671579418602, 758481182157460, -201531065820906, -778793817550197, 150472617351805, 320445662317190, -62896893877735, -61372297713001, 12968704565682, 5328164864492, -1202197738381, -161274827528, 37752739446], [123665771390928, -170600783335470, -842622003635849, 244176912750701, 844077466448136, -178318841583876, -338261095735141, 70296111977863, 63333380868157, -13841584361987, -5401712538693, 1243644226300, 161631948769, -38350226953], [106272852513750, -128265388501436, -754127045359874, 141244421821633, 790930436132310, -113117298126910, -332421010230685, 54689127249561, 64701425405268, -12367234402564, -5680374898316, 1207022732481, 173061237061, -38926641911], [150684116159928, -184286450571781, -1049938391405821, 184820997223490, 1069606585109949, -140131255986980, -438307828557386, 67191645924951, 83843136303042, -15243485778272, -7282180176589, 1494088890444, 220483599101, -48334159531], [295489137970695, -392875051320628, -2084788718214012, 441253708065913, 2151287002908579, -330103734664163, -890755378335540, 149960354490979, 171744662126800, -32990628819313, -15003756789322, 3186087569260, 456007662555, -102355805795]];
    Rf_basisdens := [1, 1, 837106466514, 837106466514, 837106466514, 837106466514, 837106466514, 837106466514, 837106466514, 837106466514, 837106466514, 837106466514, 837106466514, 837106466514];
    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, 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, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0], [-1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 1, 0], [-2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [-1, -1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, -1, 0, 1, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 1, -1, 0, -1, 0, 0], [0, -1, 1, -1, 1, 0, 1, 0, 0, 0, -1, 0, -1, 1], [0, 1, -1, 1, 0, -1, -1, 0, 0, 1, 1, 0, 1, 0], [-2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1], [-1, 0, 1, 0, 0, 1, 0, -1, 0, -1, 0, -1, 0, 0], [-1, -1, 1, -1, 1, 0, 0, 1, 0, -2, 0, -1, 1, 0], [2, 0, 1, 1, 1, 0, -1, -1, 2, 0, 1, 0, 1, -1], [2, -1, -1, 0, -1, 0, 0, 1, 0, -1, 2, -1, 1, -1], [-3, 0, 1, 0, 1, -2, -1, -1, 0, 2, -1, 0, -1, 0], [1, 1, 2, 0, 0, -1, 0, 0, 1, 1, 0, 0, 1, -1], [4, -1, 1, 1, 1, -1, 0, -1, 1, 1, 0, 1, -1, 0], [1, -1, -2, 0, -1, 0, 0, 0, -2, 1, 0, 0, -2, 2], [0, 0, 0, -1, 1, -1, 2, 1, -1, 0, -1, 0, -2, 1], [1, -2, -1, 0, 1, -2, -1, 0, -1, 0, -1, 1, -2, 1], [0, 0, 0, 0, 1, 0, 1, 0, 2, 3, -1, 1, -2, 1], [-1, 0, 1, 0, 0, -1, -1, -1, 0, 0, -1, -1, 0, -1], [-3, 1, 1, 0, -1, 1, 0, 0, 2, 0, 2, -1, 2, 0], [5, -2, -2, 0, -1, 0, -1, 1, 0, -2, 2, 1, 1, 0], [1, 0, -2, 1, 0, 0, 2, -1, -1, 2, -2, 1, -2, 1], [2, 0, 1, -1, 0, -1, 1, 0, 0, 0, -1, 1, 0, -1], [-1, 1, 0, 0, 0, 0, -1, 0, 0, 0, 1, -1, 3, -2], [5, -3, -1, 1, 0, 0, 1, -1, -2, 1, -1, 2, -3, 0], [5, -2, -1, -1, -1, 1, 1, 0, -2, 0, -3, 0, -2, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, -1, -1, -1, -1, 1, 1, -1, 0, -1, 2, 0, 1], [-2, -3, 2, 0, 1, -1, -1, 1, 1, 1, 0, -1, 0, -1], [1, 2, 0, 1, 1, -1, 0, 0, 2, 1, 0, 1, 0, 0], [0, 0, -1, 0, -1, -1, -1, 0, -2, -2, 1, 2, -1, 0], [2, -2, 1, 0, -1, 0, -1, 0, 1, -2, 1, -1, 1, -2], [-1, 1, -1, -1, -2, 1, 1, 3, 0, -1, 2, -1, 2, -1], [6, -2, 1, 1, 2, 0, 0, -2, -1, 3, -2, 2, -2, 1], [2, -1, 2, 0, 0, 1, 1, 1, 2, 0, 1, -2, 2, -1], [0, -2, 1, 1, 0, 0, 0, -2, 1, 0, 0, 0, -1, 2], [-8, 2, -1, 1, 3, 0, -1, 0, 1, 1, -1, -1, 1, 1], [2, 1, 1, 0, -1, 0, -4, 0, 3, -2, 4, 0, 4, -2], [7, 0, -1, 0, -1, 1, 0, 2, -1, -3, 4, -1, 4, -3], [4, 0, 0, -1, 2, -1, 2, 0, -1, 2, -2, 2, -3, 2], [5, -2, -2, 0, 0, 1, 2, 2, -1, 1, 0, 0, 1, 1], [-4, -1, 1, 2, 2, 0, -1, -2, 3, 2, -1, -1, -1, 1], [6, 0, -1, 1, -1, 1, 1, -2, -1, 0, 2, 2, 0, 0], [-3, 0, -1, 1, 1, -1, 2, 0, -1, -1, -3, 1, -2, 2], [-2, 1, -2, 1, 0, 0, -1, 2, 0, 0, -1, 1, 2, 1], [-3, 0, 1, 0, 1, 0, -2, -2, 2, 0, -1, -1, -1, 1], [-1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, -1, 1], [3, -1, 0, -1, 1, -1, 0, -2, -1, -1, -2, 0, 0, 1], [3, 3, 1, 1, 0, 0, -1, -1, 0, 0, -1, 0, -1, 1], [2, 3, 1, -2, 0, 0, 1, -1, -2, -1, -1, 0, 0, 2], [0, 0, -1, -1, 2, -1, 2, 1, 0, 2, -2, 1, -3, 2], [2, 2, 1, -2, 0, -1, 2, -1, 1, 3, -2, 3, -2, 2], [-2, -1, 1, 0, 3, -3, -2, 0, -1, -2, -1, -1, 1, -2], [-2, 2, 1, 0, 1, -2, 2, -2, 0, 2, 0, 1, -1, 2], [1, 0, 1, -1, 2, -3, -2, 1, 3, -2, 2, -3, 2, -1], [0, 0, 3, 1, 1, -1, -1, 0, 1, 1, 0, 2, -1, -2], [8, 2, -3, 2, -1, -1, 1, -2, -2, 5, -1, 3, -3, 2], [4, 1, -2, 3, 0, 0, 0, -1, -1, 3, -2, 2, 0, 1], [2, 2, 2, 1, 0, 1, -2, -2, 3, -2, 0, 0, 3, -2], [-7, 2, -2, 0, -2, 2, 2, 1, 0, 4, -1, 0, -2, 1], [-4, 2, 2, 0, 0, 3, 2, 3, 1, -1, 1, -4, 2, -3], [-1, 2, 3, -1, 0, -2, 1, -1, 3, 2, -1, 1, 0, -2], [7, 1, -1, -1, -2, 0, -2, 0, -1, 0, 2, 3, 3, -1], [6, 2, -3, 0, -3, 3, 3, 1, -3, -2, -1, 0, 1, 0], [-3, 2, -2, 0, 1, 0, 1, 0, 1, 4, -1, 3, 1, 3], [4, -3, -1, 1, 2, 0, 1, -1, -4, -3, -3, 2, -1, -1], [0, 1, 2, 1, 2, -2, -1, -1, 1, 4, -1, 0, -1, 1], [-4, 2, 0, -1, -2, 3, 2, -1, -2, -1, -2, 3, -2, 1], [8, 1, -1, -1, -2, 3, 1, -1, -1, -2, -3, 2, 0, 2], [7, 2, -4, 0, -3, 3, 1, 0, -3, -3, 1, -2, 1, 0], [0, 3, -3, 0, 0, 0, -1, -2, -1, -3, -1, 1, 1, 2], [-2, 2, 2, 0, -3, 2, -1, 0, 2, 0, 3, -1, 3, -3], [-2, 1, -1, -1, 1, 4, 2, 5, -1, -5, 0, -3, 1, 1], [0, 1, 1, 2, -2, 2, -3, -2, 3, 1, 1, -1, 3, -2], [4, 1, 0, -3, -1, 0, 2, 0, 0, 3, 0, 2, -2, 2], [-3, 4, 4, -1, 1, -2, -1, -2, 4, 3, 3, -1, 1, 0], [-3, 5, -2, -1, -1, 1, 2, 1, 2, 4, -2, 1, 0, 1], [-7, 1, 1, -2, 2, -2, -1, 2, 3, -4, -1, -5, 2, 2], [3, 0, -1, 0, -1, -1, -3, 2, 0, -2, 3, -1, 2, 2], [-1, 1, -2, 0, 1, 0, -2, 0, 0, 1, 0, -4, -4, 2], [8, -1, 3, -1, 3, -1, 0, -2, 0, -3, 0, -2, 3, -1], [2, -2, 4, -1, 2, -1, 3, -1, 1, 3, 0, 1, -5, 3], [2, -4, -1, 2, 2, 1, 1, 0, 0, 1, -1, 2, 0, -1], [-2, 3, 0, -2, -1, -2, 0, 4, 2, -3, 4, -2, 2, 0], [-1, 1, 5, 1, 5, -4, -2, -1, 4, 1, 2, -5, 2, -2], [-6, 4, -2, 0, -1, -2, -3, 1, 3, 2, 5, 4, 3, 0], [-2, 2, 1, -2, 1, -1, 4, 3, -2, 3, -3, 2, -6, 3], [-1, 0, -2, -2, 0, 0, 1, -2, -2, 5, -5, -1, -6, 4], [-3, 0, 0, 1, 1, 0, -1, 0, -2, -2, 1, -3, 3, 1], [-1, 0, 3, 1, 0, 1, 0, -1, 7, 4, 2, -2, 1, 1], [2, 0, 1, 1, 3, -2, 3, 1, 4, 2, -1, 2, -1, 3], [6, -1, -1, -1, -2, 1, 1, 1, -3, -1, -3, -2, -1, -2], [9, 0, 0, -1, -4, 2, -2, -1, -5, -3, 2, -3, 0, -2], [-4, 0, 3, -2, 5, -1, -1, -1, -1, 2, -5, 0, -2, 0], [0, -6, 4, 3, 1, 1, -1, 0, 1, 1, -1, 0, -1, -3], [13, 0, -2, 1, -3, 3, -1, 0, 0, -2, 3, 1, 3, 0], [3, 2, -1, 3, -4, 1, -2, 0, 2, 0, 4, 4, 0, -2], [6, -3, 4, -4, 4, 0, -1, 3, 4, -4, 1, -4, 4, 2], [1, 0, 3, -2, 2, -2, 2, 1, -2, 1, -2, 0, -2, 0], [-1, -2, 0, 0, 0, 1, 2, 4, 3, 0, 2, -1, -1, 2], [-2, -2, -1, 3, 2, 0, 1, -1, 4, 1, 1, -1, 1, -4], [-4, 4, 3, -1, 0, 1, 0, -3, 2, 0, 2, -1, 3, 2], [-5, -1, 0, -3, 2, -1, 5, 1, -2, -2, -5, 2, -5, 4], [-1, -4, 3, -2, 3, 0, -1, 1, 8, -4, 1, 2, 6, 0], [8, -2, 3, 1, -2, 2, 0, -1, 0, 1, -2, 0, -5, -1], [-4, 2, -1, 2, -3, -1, -2, 3, 1, -1, 4, -4, 4, -2], [7, 3, -8, 0, -2, 2, -1, 3, -1, 3, 1, -1, 2, 2], [13, 2, -3, -2, -1, 2, 3, 1, -4, 0, -1, 2, -2, 2], [-6, -2, 1, 0, 5, 1, -1, 0, 2, 2, -1, -5, 0, 0], [7, -2, 1, -3, -1, 2, 1, -1, -4, -4, 1, -2, -1, 1], [-2, 3, -1, -3, -3, 1, 0, -1, -3, -2, 0, 1, 2, 2], [9, 0, -2, 1, 2, 0, 1, -4, -5, -1, -2, 4, -5, 1], [1, 0, 1, -2, 1, -1, 1, 3, 3, -6, 5, -2, 3, -2], [0, 0, -1, -2, 2, 1, 4, -4, -4, 0, -2, 0, -3, 3], [11, -6, -2, 3, -2, -2, -4, 0, 0, 0, 4, 3, -3, -2], [3, -3, 0, 4, 1, 2, 0, -1, -1, 0, -4, -1, -2, -2], [1, -2, 2, -2, 2, -1, -2, 3, 6, -5, 2, 1, 9, -2], [4, -1, -3, -1, 0, 1, 1, 7, -3, -6, 1, -2, 2, 0], [11, -1, 1, -2, -2, 3, 3, 1, -3, -1, -1, 2, 0, 1], [0, 7, -3, 3, 0, -1, 1, -5, -5, 5, -3, 4, -4, 3], [-4, 1, 3, 3, 4, 1, 1, 0, 2, -2, -1, -3, 1, -2], [-2, -6, 0, -2, -2, 2, 0, 2, -2, -7, -2, -2, 1, 1], [17, 2, -3, 0, 2, 2, 2, 3, 0, -3, -2, -3, 5, 0], [-8, 0, 2, 1, 4, -5, 1, 0, -2, 3, -4, -4, -2, -1], [4, -5, 2, -5, -2, 4, 1, 3, 1, -5, 1, 2, 2, 0], [6, -1, -1, -2, -3, 4, 0, -3, 0, 0, -2, 3, 2, 0], [7, 3, -1, 1, 0, -2, -5, -3, 0, 0, 3, 2, 9, -2], [1, -3, -3, 3, 1, -2, 0, -1, 2, 4, 0, 4, -1, 1], [13, -3, -3, 2, 0, 0, 1, 1, -3, 2, -2, 1, 0, -3], [-11, 0, 0, 0, 2, -1, -2, -1, -2, 2, -2, 1, 3, 3], [-2, 0, -1, 6, 1, -1, -5, -2, 4, 4, 1, 3, -2, 0], [12, -5, 3, 0, -1, 2, 2, -1, 0, -3, -4, 1, -3, -1], [6, -1, 2, -3, 3, 1, 3, 0, 1, 1, -3, -1, -1, 3], [9, 0, 5, -2, 1, -2, -4, -1, 3, -1, 1, 4, 2, 0], [-16, 3, 3, 0, 3, -4, 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5, -2, -2, 4, 1, 7, -7, 3], [-20, 6, 0, 2, 1, 0, 3, -5, -1, 8, -3, 2, -6, 1], [-4, -4, 3, 3, 5, 2, 1, 2, 3, -3, -1, -5, 4, 5], [0, -3, -9, 4, -1, 0, 2, 0, -3, 4, -2, 6, -7, 3], [-8, -5, 2, 6, 1, 1, -2, -1, 0, 0, 4, -3, -2, -3], [-9, -2, 2, -2, 9, 3, 3, 3, 6, -2, -1, -5, 1, 2], [8, -4, -1, 3, 4, 2, -1, -4, -5, -4, -3, -6, 0, -3], [14, -8, -1, -1, 1, 2, -1, 1, -6, -8, -3, 6, -1, -1], [2, -10, 0, -2, 1, -4, 3, -3, -1, 7, -5, 6, -7, 3], [6, 0, -1, 2, 8, -3, 2, -5, -3, 7, -2, 0, -4, 3], [-6, -10, 9, 0, 5, 0, 1, -1, 1, -1, -8, -9, -2, -1], [-13, 7, -1, 4, -5, -2, 0, 2, -2, 1, 0, 2, -11, -2], [-5, 2, -3, -3, -3, -5, -5, -5, 5, 8, -1, 2, -1, 0], [5, 1, 6, -1, 0, 6, 1, 1, 2, -5, 3, -8, 13, -4], [2, 14, -1, -2, -4, 3, 2, 1, -3, -1, 0, 5, 2, 2], [1, 0, 5, -4, 2, -2, 0, -4, 1, 11, -4, 2, -8, 4], [-13, 1, -3, 4, -2, 1, 3, 4, -4, -3, 3, -1, 2, -2], [3, -7, 1, -4, 6, -1, 5, 0, -2, -2, -13, 3, -8, 8], [-5, -6, -4, 3, -4, 5, 1, -3, -2, -2, -3, 2, 2, 3], [-5, -8, -5, -1, 4, 0, 1, 3, -3, 1, -3, 5, -1, 4], [-3, 4, 9, -6, -4, -6, 0, 5, 0, 2, 0, -1, -2, -4], [11, -10, -1, 1, 1, 5, 1, -6, -8, -9, -5, -3, 7, -2], [-5, -3, -6, 2, 2, -2, -1, -1, 1, -4, 1, -4, 7, 5], [-15, 4, 2, 5, 4, 2, -2, -4, 2, 0, 2, 4, -1, -1], [12, -6, 1, -5, -3, 6, 1, 5, 2, -5, 1, -5, 8, -2], [-19, 4, 2, -8, 3, -1, 5, 2, 1, -1, -5, -3, -2, 3], [-6, -2, 7, -1, -7, 4, 3, 1, 2, -7, 1, 2, 2, -5], [-22, 2, 4, -4, 5, -5, 3, 3, -1, -2, -3, -10, -2, 0], [-12, -7, 8, -2, 10, -9, -1, -7, 4, 1, -3, 4, -2, 4], [-13, 1, -8, 5, -3, 3, 4, 6, 1, 9, 0, 2, -2, -3], [-23, -1, -4, 0, -4, -2, -2, 2, 1, 1, 5, 13, -6, 3], [-25, 5, -3, 2, -3, 0, 1, -1, -3, 9, -1, 7, -18, 4], [10, -6, 4, 3, 1, -3, -1, -7, -2, 0, -5, -4, -10, -2], [1, -4, -8, 6, -3, 1, 1, 2, -1, 1, 1, 2, 4, -2], [-10, -3, 0, 1, -2, 2, -1, -3, 7, 2, 1, -4, 4, -2], [13, -3, 9, 1, -3, 8, 2, -1, 4, -3, -4, -5, -5, -1], [-7, -4, 1, -4, -2, 0, 6, -1, -2, -3, -4, 5, -3, 2], [-2, -5, 4, -4, -2, 1, -1, 3, 4, -7, -7, 4, 0, 0], [-9, 3, -1, 2, 3, -3, -8, 4, 6, -9, 10, -8, 9, -5], [-8, 8, 4, -1, 2, -2, -8, 5, 7, 0, 4, -9, 4, -1], [-14, -6, -3, 3, -3, -2, 1, -4, -3, 11, -1, 5, -8, 8], [11, 7, 3, -7, -3, 2, 6, -6, -9, -2, -6, 2, -6, 4], [-1, -4, 0, -2, -2, 7, 2, 4, -5, -4, 2, 4, -3, -6], [-2, -3, -4, -4, -5, -3, -4, -3, -2, 2, 0, 9, -2, 5], [5, 2, 2, -9, -3, 4, 9, 3, -5, -2, -5, -2, -2, -3], [-19, 10, -3, 2, 2, 2, 2, 0, 3, 4, -6, 2, 0, 1], [-7, 0, 1, 4, -1, 0, -3, 2, 7, -1, 7, 0, -2, -4], [-7, -11, -2, -1, 3, -1, -2, 0, -3, -6, -2, -8, 8, 1], [21, -13, 3, -1, -2, 2, 7, 0, -5, -8, -1, 1, -8, 0], [-6, 6, 1, -2, -1, 0, 4, -6, -2, -5, -2, 3, 4, -2], [-14, 7, 10, 1, 3, -4, 1, -4, -1, 3, -2, 0, -6, -5], [4, 8, 3, -2, -7, -1, -7, 0, 2, -2, 5, -5, 7, -5], [7, 5, -9, 5, -5, 2, 0, 1, -2, -6, 0, 4, 7, -7], [-7, -4, 0, 3, 3, -2, 5, -3, -3, 7, -9, -2, -6, 2], [19, 0, -4, -1, 0, 0, -4, 7, 1, -6, 14, -7, 10, -5], [-14, 2, 5, -2, 11, -7, -5, -8, 8, 9, -5, 7, -4, 5], [14, -3, -6, 0, -4, 6, 3, 3, -6, -12, -1, 0, 4, 0], [30, -6, -1, 0, -4, 0, 3, -1, -3, -5, 5, 3, -7, -4], [10, 7, -3, 6, -4, 10, 1, -1, -6, 0, 5, 3, 3, -2], [-17, 4, -1, 6, -2, -4, -2, -5, 4, 9, 6, 3, -4, -3], [-16, -1, 6, 8, 6, -2, -4, -1, 9, 3, 4, -3, -3, -1], [15, -8, 7, -3, -2, -1, 1, -4, -4, -12, 1, 0, 2, -4], [-3, -3, -6, 1, 7, 1, -2, -2, -3, 1, -4, -10, 5, 2], [-2, 2, 0, -6, -3, 0, -4, 2, -2, -1, 0, -1, 0, -3], [-15, 2, 1, 8, 3, 0, -7, 0, 3, -2, 1, -3, 10, -1], [-9, 2, -3, -2, -2, 0, 0, 7, 4, 1, 4, -4, 8, -2], [-8, -15, 3, -2, 3, -3, 2, 5, 2, -8, 2, -1, 1, 1], [-1, -1, -5, 5, 3, -2, 2, -2, 5, 0, -4, 6, 3, -3], [15, -3, -3, 7, -10, 2, -7, -8, 1, 1, 7, 1, 9, -2], [-27, -3, 6, 3, 1, -3, -2, 0, 1, -2, 2, 4, -1, -5], [3, 7, -6, 0, -10, -1, -7, -3, -1, 1, 3, 8, 6, -2], [4, -3, 6, 0, -5, -4, -3, 3, 0, -3, 7, 3, 1, -4], [1, 8, 0, 1, 9, 4, 9, 3, 1, 6, -9, -2, -3, 4], [-8, -1, 6, 0, 0, -3, -2, -7, 0, 9, -4, 8, -7, 3], [-10, 2, 2, 2, 0, -8, -8, 1, 1, 3, 6, -11, 6, -2], [6, 9, -6, 2, -5, 1, 0, 5, -2, 2, 2, -3, 5, -2], [8, -8, 4, 0, 3, 8, 7, 3, -3, -10, -1, -3, 4, -6], [-14, 0, -1, 8, -3, -3, -9, -2, 8, -5, 9, -3, 11, -10], [15, 10, 1, -1, -4, 3, -4, 7, 5, -11, 8, -1, 20, -2], [6, 9, -7, 1, -6, 3, -1, 1, -5, -1, 9, -4, 14, -9], [12, -10, 5, -1, 2, 0, 0, 7, 4, -1, -2, -1, 1, -2], [3, 4, 0, 3, -7, 3, 0, -2, -1, 7, 4, 6, 0, -5], [-16, 0, 5, -2, 5, -3, 5, -1, 1, -1, -5, 8, -8, 2], [-8, 5, -10, 8, -8, -7, -9, -4, 1, 6, 7, 7, 1, -7], [-6, 4, 2, -4, 1, 0, 0, -3, 3, 1, -4, 0, 0, 2], [-6, 10, -5, -10, 3, -5, 1, 0, 2, -1, -5, -3, 5, 4], [-3, 0, -10, -2, -5, 5, -1, 6, -1, -5, 5, -3, 12, 0], [-13, -3, 1, -4, 1, 4, 13, 1, -9, 8, -9, 3, -12, 7], [22, 5, 2, -1, 2, -6, 3, 3, 5, 8, 3, 2, -3, -1], [2, -13, 3, 1, 4, -5, -1, -1, -1, -2, 5, -2, -6, 0], [1, 6, 4, -7, 5, -2, 5, 3, 1, 9, -3, 4, -8, 2], [-15, 8, 1, 1, -9, 5, -1, -3, 3, -6, 7, 9, 8, -3], [-5, -5, -4, -3, 1, -2, 0, 7, -1, 6, -2, -1, -5, 3], [13, 9, -2, -5, -6, 5, 1, 0, 1, -1, 3, 5, 5, 2], [-23, 4, -8, 6, 1, 3, 3, 1, -8, -3, -5, 4, 3, 2], [-17, -5, -7, 2, 1, 5, 2, 4, -2, 0, 0, -4, 5, -3], [16, 5, -6, 0, -2, -5, -5, 4, -5, 2, 3, 3, 1, -2], [-2, -5, 4, 2, -3, -5, -2, -6, -1, 9, -2, 10, -13, -2], [20, 1, -7, 0, -4, 2, -1, 2, -3, 5, 1, 1, 4, -3], [-16, 10, 3, -5, 6, -6, 0, 0, -1, 2, -2, 0, -3, -2], [22, -3, -4, 4, 3, 0, 7, -6, -11, 6, -9, 8, -10, 3], [8, 2, -13, 0, 0, 1, -2, 0, -10, -1, 2, 4, -2, 5], [-3, -3, 6, 2, 5, -6, -4, -2, 6, 1, 0, 4, -10, 9], [9, -6, 5, 5, -2, 5, 2, -6, 8, -3, 0, 0, 4, -5], [-15, -3, 4, 2, 13, -6, -8, 2, 8, -6, 2, -4, 4, 0], [4, 9, 0, 0, -4, 3, -7, -1, 2, -4, 1, 2, 7, 0], [-17, -1, 4, -1, 3, -3, -2, -3, 4, 0, 0, 1, 4, 5], [-29, 12, -3, 1, -1, -5, -5, -9, -1, 10, -1, 4, -4, 5], [-8, 7, -1, 2, 0, -4, -5, -4, 7, 2, 3, 3, 6, -4], [-16, -1, 4, -3, -5, 4, 0, 6, 6, -3, 4, -4, 8, -6], [20, 2, -5, 4, 7, -1, 2, -5, 2, 15, -3, 2, -4, 11], [-10, 4, 3, -1, 9, -4, -1, 1, 6, 6, 1, 6, 1, 1], [-16, 10, -4, -1, 1, -3, -7, 4, 3, -5, 9, -4, 6, -4], [-2, 8, 1, -6, 0, 5, -2, -2, -2, 3, 2, 0, -5, -2], [9, 1, 1, -2, -1, 3, 0, -1, 7, 0, 0, 5, 0, 4], [21, -7, 5, -3, 3, -1, 1, 1, -6, -3, -5, -5, 6, -3], [-3, -8, 2, 0, 2, -6, -1, 2, 3, 5, -3, 2, -5, -2], [26, -12, 3, 1, -3, 4, -3, 0, -3, -11, 5, -3, 10, -10], [20, 2, 7, 2, 8, -7, -6, -5, 5, -4, 4, -1, 10, -1], [1, 0, 2, 3, 1, 0, -3, -3, 5, -2, -1, -4, 9, -8], [-13, 5, -10, 0, 0, 0, -1, 8, 2, 5, 5, -3, 6, 2], [13, -1, -3, -2, -5, 5, -1, 2, 7, 5, 6, 0, 1, 4], [9, 5, 3, -1, 1, -7, -3, 3, 5, 4, 9, 2, -1, 0], [4, 1, 2, -6, 2, -7, -7, -2, -3, -1, 3, 5, 6, 1], [-6, -4, -4, 1, -8, -9, -7, 0, -1, -6, 15, 10, 2, -3], [9, -6, 2, -1, 5, -4, 1, 4, 0, 0, 5, -3, -2, 0], [36, -2, 0, -6, 1, 4, -1, 4, 0, -11, 3, -2, 11, -1], [-3, -9, 2, -7, 7, -3, 4, -3, -2, 5, -8, 7, -13, 7], [-4, 3, 0, 3, 0, 2, -3, 0, 2, -10, -3, -5, 4, -5], [-41, -3, 1, 0, -1, -5, 4, -3, 3, 10, -2, 9, -8, 0], [10, 14, 0, 0, -11, 8, 1, 2, 2, -8, 7, 1, 5, -3], [12, 6, 0, 1, 3, -8, -3, -5, -4, 15, 0, 1, -2, -6], [9, 5, 0, -5, -2, 3, 3, -3, -2, 11, 5, 2, -6, 0], [4, -8, 5, -4, -1, -5, -5, 0, 4, -5, 5, -1, -5, 2], [-12, 13, -7, -5, -2, 2, 8, 7, 1, -1, 0, -4, -3, 1], [4, 6, 1, -8, -2, 3, 5, -4, -7, -4, -2, -6, -4, 1], [-11, 1, 2, 3, 3, -1, 8, 4, -3, -1, -4, -12, -9, 1], [-9, 5, 8, 5, 8, -1, -1, -5, 2, 0, -1, 0, -7, 4], [-2, 3, -7, -3, -7, -6, 8, 3, -10, 7, -4, 11, -10, 2], [17, -11, -6, 0, -5, 3, 5, -1, -2, 7, -7, 10, -7, -2], [-17, 0, 9, -11, 6, -1, -2, 4, 4, -6, -2, -8, 2, 2], [-14, -9, 9, 4, 3, -4, -8, -3, 8, -3, 2, -1, 8, -10], [-11, 3, -1, -2, 5, -3, -2, -3, 1, 6, 2, 1, 2, 2], [-3, 3, 4, 2, 3, -6, -2, 6, 0, 4, 2, -4, -2, -1], [4, -12, 12, -2, 4, 0, 2, -3, 1, -5, -2, -5, 1, -3], [-25, 0, 7, 2, -2, -2, 0, -2, 7, 4, 0, 6, 0, -5], [-13, -2, -1, 0, 4, 0, -3, 1, 8, -1, 7, 7, 3, -4], [-6, -1, -2, 1, 7, -9, -3, -3, 0, 3, -3, -2, 3, 8], [-6, 1, -3, 0, -4, -2, -7, 0, -3, 13, -5, 1, -8, 1], [14, -4, -1, -4, -8, 9, 10, 7, -1, 5, -4, 3, -4, 4]];
    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_q();".
// 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_q(:prec:=14)
    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_q();".
// 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_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8046_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-7083, -6282, 53551, -9808, -55580, 12990, 24455, -5066, -5559, 882, 680, -70, -42, 2, 1]>,<11,R![-331776, -1124352, 2629632, 315840, -1875456, 104152, 476189, -37906, -56147, 3815, 3294, -150, -93, 2, 1]>],Snew);
    return Vf;
end function;