
// Make newform 6045.2.a.bg in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6045_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_6045_2_a_bg();".
// 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_6045_2_a_bg();".
// 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 := [-428, 2178, -780, -8572, 6027, 11579, -8358, -7583, 5124, 2678, -1657, -517, 294, 51, -27, -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, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -6, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2656763704, -7260966308, -5347435986, 22883291190, -891629561, -24399073216, 5927851406, 11616670989, -3840982829, -2652072475, 1041027406, 266535327, -127878505, -6105068, 5831475, -401375], [-2433705296, 5350743020, 9022573940, -16197805786, -10588021223, 17484344834, 5448761876, -8903486735, -1333309911, 2381186265, 144806888, -340606193, -4276819, 24375904, -166555, -675635], [3930275208, -13136412272, -9364329458, 41723176503, -3092968523, -42361156993, 14581580542, 18411647961, -9616165022, -3443080522, 2652036733, 146821522, -329766812, 28420427, 15164660, -2488534], [1615863693, -5053913356, -3814744471, 14233568984, -998374179, -12986541279, 5408123199, 4843358589, -3598060725, -579282729, 997145395, -74325399, -124455102, 22545964, 5740111, -1368167], [-1177545982, 8548552388, -8620241002, -31297948725, 32323253840, 33292767751, -34346738054, -14429646670, 16331960235, 2468886049, -3861029905, -11139427, 439852093, -38546041, -19153621, 2803945], [6127161078, -13542412620, -25301350906, 44928219971, 31714926198, -50523613135, -18150614914, 26756629056, 5399648587, -7510769565, -852753033, 1147216629, 67211901, -89938815, -2064367, 2831059], [-4318069210, 10687542066, 16776685232, -35772605869, -21686057951, 41339037949, 13556760324, -22628667487, -4600819186, 6623436958, 859595697, -1067461518, -82563440, 89382371, 3168600, -3037054], [-1296448684, 7707270874, -8447540368, -23543309537, 36425878673, 18923186773, -41288117178, -3214863019, 20406762218, -1961321472, -4945595009, 914873352, 572705046, -135487993, -25215264, 6782934], [-2141597206, 1757645388, 19765596698, -5136433166, -47064899461, 10655400508, 44968494856, -10968126505, -20586373127, 5456991977, 4782945210, -1356968093, -539774147, 161207194, 23370511, -7250209], [1789943440, 48314368, -22956255820, -1026742203, 59141515631, -6027776741, -58610211646, 10688873685, 27427624976, -6300243366, -6462936649, 1686862550, 736125094, -208198863, -32068302, 9572188], [-4303214968, 18415371574, -3826053620, -59309144451, 46082550133, 55300353944, -59479946487, -18471962304, 30983285268, 543799535, -7729231159, 923369914, 911381417, -177132377, -40614744, 9720317], [-2285385432, -1865853208, 29046054534, 4251737464, -76558797477, 7413524609, 76333936461, -14882591054, -35648126769, 8718170552, 8361431816, -2295935223, -947900454, 279181880, 41128413, -12686056]];
    Rf_basisdens := [1, 1, 1, 1, 66641158, 66641158, 66641158, 33320579, 66641158, 66641158, 66641158, 66641158, 66641158, 66641158, 66641158, 66641158];
    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_6045_a();"
function MakeCharacter_6045_a()
    N := 6045;
    order := 1;
    char_gens := [4031, 4837, 1861, 2731];
    v := [1, 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_6045_a_Hecke(Kf)
    return MakeCharacter_6045_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], [-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, 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, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0], [0, -1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1], [-1, -1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, -1, 1, 0], [0, -1, 0, 0, 1, 1, 0, 1, 0, 1, 0, -1, 0, 0, 1, 1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, -1, -2, 0, -1, 0, 0, 0, -1, -1, 1, 0, 0, -1], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 1, 0, 1, -1, -1], [-1, 0, 0, 1, 1, 1, 1, -1, 0, 1, 2, 0, -1, 0, -1, 0], [1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 1, -1, -1, 0, 0, 1], [-3, 0, 1, 0, 2, 0, 2, -1, 0, 1, 2, 1, -1, -1, 1, 1], [3, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, 0, 0, 1, -1, 0], [2, -1, 2, -1, 0, 0, 1, -1, 1, 0, 0, -1, 0, 0, 1, -1], [0, 0, 1, -1, 0, -1, -1, 0, 0, 0, -1, -1, -1, 1, -1, 0], [-2, 0, -1, 1, 1, 1, 0, 0, 1, 0, -1, 0, 2, 0, 0, 0], [1, 0, -1, 1, 1, -1, 1, -1, -1, -1, 0, 1, 1, 1, -2, 1], [-1, -1, 1, -2, -1, 0, 0, -1, 2, 0, 0, -3, 0, 1, 2, 0], [-1, -1, -3, 0, -2, 1, 0, 1, 0, -2, -1, 1, 1, 0, -1, -2], [0, 0, 1, 0, 1, 2, 0, 0, 0, 1, 2, -1, 0, -1, 0, -1], [1, -1, 0, 0, 2, 1, 0, 0, 0, 0, 2, -1, -1, 0, -2, 1], [1, 2, -3, 2, -1, 0, -1, 2, 1, -1, 0, 1, 0, -1, -2, 1], [-1, -2, 3, -1, 0, 1, 0, 1, 0, 1, -3, 1, 3, 0, 2, -2], [2, -1, 0, 0, -3, -1, -2, 0, 1, -2, -3, 2, 3, -1, -1, -3], [1, -4, -1, 0, 2, -1, 1, 1, 0, 1, 2, 0, -1, 1, -1, 3], [-2, -2, 1, 2, 2, -1, 1, -2, 0, 1, 1, 2, -1, 1, -2, 2], [2, 1, 0, 0, 0, 1, 1, 3, 0, 1, 0, -1, 0, -1, 3, 0], [5, 0, 0, 0, 0, -1, 0, 0, 0, -1, 1, 0, -4, -1, -2, 0], [-3, 0, -2, 0, 2, -1, 0, -1, 0, 0, 2, 0, -2, 0, 0, 1], [3, -2, 1, 1, 0, -2, 1, -1, 1, 0, 0, 0, -1, -2, 0, 3], [-1, 0, -1, 1, 1, 0, -1, -1, 1, -2, 0, 2, 1, 1, -2, 0], [-2, 2, -2, 0, -3, 1, 0, 1, 1, -1, -3, 2, 3, 0, -1, -3], [-1, 2, -1, 1, 2, 1, 2, -1, -2, 1, 1, 1, 1, 0, -2, -1], [-4, 4, -1, 2, -2, -3, 0, -4, 0, -2, -1, -1, -1, 0, -4, 0], [-2, -2, 0, 0, 0, 1, 0, 1, 1, 0, -2, 0, 1, 0, 2, 1], [0, -1, 3, 0, 1, 2, -1, 2, 0, 1, 1, -2, -1, -2, 3, -2], [2, 3, -3, 2, 0, 2, 0, -2, 0, -2, -2, 1, 0, 1, -3, -2], [6, 1, 2, 2, 2, 1, 0, -1, 0, 1, 1, 1, -1, -2, -1, -1], [-3, 3, -1, 2, -2, 1, -2, -2, 0, 0, -2, -2, 0, 0, -1, -1], [2, 0, -1, 1, 2, 2, 1, 1, -1, 0, 4, 2, -2, 1, -2, 0], [-1, 4, -1, 0, -3, 0, -2, 0, 0, -3, -4, 2, 3, 1, -2, -1], [4, -3, 2, -2, 0, 1, 1, 1, -1, 1, 0, 0, 1, -1, 4, -1], [1, -1, -1, 0, 0, 1, 0, -1, 2, 1, 3, -5, -1, 0, 3, 1], [-3, 0, -3, 0, 1, 1, 0, 3, 0, 0, -1, 2, 2, 3, 1, 1], [0, -1, -2, -2, 1, 1, 1, 1, -1, 0, 2, -1, 0, 3, 0, -1], [4, -4, 1, -1, 0, 1, -1, 2, 1, 1, 1, -2, 2, -1, 2, 0], [-2, 1, 1, 2, 2, -1, 0, -3, -1, 2, 4, -1, -5, -1, -3, 2], [0, 0, 0, 0, 0, -1, 0, -2, -1, -4, -1, 2, -2, -1, -3, -1], [0, -2, -1, -1, 2, -2, 1, -3, -1, 1, 1, -1, 0, 3, -1, 1], [8, -2, -2, 1, -1, 0, -2, 4, 0, 0, -1, 3, 1, -2, -1, -2], [-5, -1, 0, 1, 1, -1, -1, 3, -1, 3, 2, -4, -1, 0, 2, 2], [-4, 2, 0, 2, 0, -2, 1, -6, 3, -2, 1, -2, -4, 0, -1, 2], [0, -2, -1, -2, -2, 0, 0, 3, 0, 2, -2, 1, 3, 2, 3, -1], [4, -2, 0, -2, 0, 3, 1, -1, 2, 2, -1, -1, 1, 1, 2, -2], [1, 1, -4, -1, -1, 0, -1, 2, 1, -1, -2, 4, 1, -1, -3, -1], [-3, 2, -2, -1, 0, 0, -1, 2, -1, -4, -4, 4, 4, 3, -4, -2], [8, 1, 1, 2, 6, 1, 2, 1, -2, 1, 3, 3, 0, -1, -1, 1], [6, -4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, -2, -2, 0, 0], [6, -4, 0, -3, 1, 1, 1, 3, -2, 4, 1, -3, 0, -1, 6, 4], [4, 3, -3, 1, -3, 0, -2, 4, -1, 1, 2, -2, -1, 1, 0, -1], [1, 4, -3, -1, -2, 0, -1, 2, 1, -1, 2, 0, 1, 0, 2, 0], [2, -5, 4, -1, -2, 1, 0, 3, 0, 4, -1, -6, -2, -2, 5, 1], [9, -5, -1, 0, -3, 1, -4, 4, -1, 2, -2, 1, 0, -3, 2, -1], [-5, 3, -2, -1, -1, 2, -1, 2, -2, 0, -1, 2, 2, -1, 1, 0], [3, -2, -2, 1, -3, 0, -2, 1, 2, -1, -3, 3, 0, -1, 2, 1], [3, -1, 3, -2, -1, -2, 0, -1, 0, 4, 3, -2, -3, -3, 4, 0], [1, 3, 3, 0, -3, -2, -4, 1, -1, 0, -5, 1, 1, -1, 2, -2], [-3, -2, 2, -1, 3, 0, 1, -3, 2, 3, 3, -5, -4, 1, 0, 3], [-5, 1, -2, -1, 0, -1, 0, -2, 1, 0, 0, -3, -2, 4, 0, 2], [0, -4, 0, -1, -1, 3, 2, 6, -1, 1, -1, -2, 6, 1, 5, 0], [2, 4, -3, 1, -5, 1, -3, -2, 1, -4, -5, 4, 4, 2, -5, -6], [-3, -1, -4, -2, -1, 0, 1, 4, -2, 0, 1, -2, 3, 4, 6, 2], [-1, 1, 1, -2, 2, 1, 3, -2, 0, 2, 3, -1, -2, 0, 0, 1], [-1, -3, 2, -2, 5, 4, 3, 2, 0, 3, 1, -4, 1, 0, 1, 0], [-2, 3, -1, 0, 1, 1, 0, -4, 3, -1, -3, 0, 0, 0, -2, -1], [0, -3, 3, -1, 4, 1, 5, -2, 2, 4, 5, -4, 0, -3, 5, 0], [11, 3, -1, 1, 3, 0, 4, 0, -1, -1, 3, 2, -2, -1, -2, 1], [6, 1, -6, -2, -5, -4, -1, 1, -1, -3, -4, 3, 3, 4, 0, 0], [-6, 0, 3, -1, 1, -4, 1, -2, -2, 2, 0, 1, 0, 0, -1, 2], [-3, -6, 0, -1, 0, -3, -2, 3, -1, 2, -2, 0, 3, 2, 4, 0], [3, -2, -1, -1, 4, -2, 1, 3, -1, 3, 1, 3, -1, 2, 2, 5], [-4, 1, 2, 2, 1, 1, 1, 0, 2, 3, 1, -4, 3, -4, 1, -1], [-4, 1, -1, 0, 0, 3, 0, 3, 1, 1, -5, 2, 6, 1, 0, -2], [3, 3, -2, -1, -4, 0, -1, 5, -1, -2, -1, 0, -1, -1, -3, 0], [-7, 3, -4, 0, -2, -1, 1, -3, 3, -1, -2, -2, 3, 0, -2, -1], [3, 2, 1, 0, 2, 1, -1, 4, -2, 2, -1, 2, 1, -2, 1, 0], [-1, 5, -2, 0, -1, -1, 0, -5, 0, 0, -3, 2, 3, 3, -3, 0], [5, 9, -1, 3, 4, 3, -1, 0, 0, -1, 4, 1, -6, -2, 0, 1], [-1, -1, 2, 0, 2, 0, 1, -3, 1, 1, 2, -1, -6, -6, 0, 1], [-4, 1, 0, -1, -1, 0, 2, -1, 2, 0, 1, -3, 2, -1, 6, -1], [2, -1, 4, 2, 3, 1, 1, 1, -2, 6, 3, -3, -2, -3, 3, 2], [3, 2, -3, 3, 1, 2, 0, -2, 1, -3, 0, 1, -3, 2, -4, 3], [5, 7, -4, -1, -9, 1, -4, 0, 3, -6, -7, 3, 5, 0, -3, -5], [5, 0, 2, -3, -2, 0, -3, 3, -3, 1, -5, 1, 2, -1, 5, -2], [1, -3, 3, -1, -3, -1, -1, 2, -2, -2, -6, 0, 5, 2, 4, -1], [3, -6, 6, -3, 1, -2, 2, -2, 3, 2, -1, -2, -3, -1, 3, 3], [8, 6, 2, 1, 2, 0, 1, -1, -1, 0, -1, 3, 3, 4, -4, -1], [-10, -3, -6, 2, 3, -2, 1, 0, 1, 0, 2, -1, 0, 2, -4, 4], [3, 0, 2, -2, -1, 0, 0, 4, 1, 0, 0, 0, 3, -1, 3, 0], [-11, 4, -2, 0, 2, -1, 4, 0, 0, 0, 2, 2, 5, 4, 2, 0], [5, -1, 1, 1, 8, 3, 6, 1, 2, 4, 6, -3, -2, -2, 3, 4], [2, -1, 1, 0, 1, 2, 1, -3, 2, 2, 0, -4, 1, -1, 4, -3], [5, 12, -1, 2, 0, -1, 2, -4, 1, -4, 2, 1, -5, 1, -5, 1], [4, 6, 3, 3, 7, 2, -1, 2, -1, 1, 8, -2, -4, 0, -1, 3], [1, 7, 1, -2, -1, 0, 3, -3, 1, 2, 1, -7, 0, -1, 5, 2], [2, -5, 4, -4, 0, 2, 2, 2, -2, 4, -1, -4, 3, 2, 7, -1], [5, 2, -3, -1, -4, -2, -1, 3, 3, -6, -6, 4, 4, 2, -3, 0], [-7, 4, -3, 3, 1, -1, -4, -2, 1, -2, 0, 1, -4, 1, -7, -1], [11, -4, 0, -1, 0, -1, 2, 3, -1, 0, 2, 7, 3, 1, -1, -1], [1, 5, -7, 2, -4, 1, 1, 0, -1, 1, 3, 0, 3, 2, 0, 0], [14, -2, 8, -4, -1, 3, 0, 5, 0, 3, 3, -3, 0, -4, 10, -1], [-5, -6, -4, -4, -2, -2, -3, 1, 3, -1, 0, -3, 2, 3, 4, 2], [8, 3, 3, 0, 2, 0, 1, -2, 2, 0, 2, -2, -7, -2, 4, 1], [6, 3, 6, 2, 6, 0, 0, -1, 1, 4, 4, -2, -8, -6, 5, 6], [-8, 5, 1, 4, 2, 3, 1, -1, -1, -2, 3, -1, -2, 2, -4, -1], [1, 6, 4, -3, -6, 3, -3, 1, 1, -2, -8, 0, 9, 4, -2, -8], [-3, 5, -3, 3, 0, 2, 3, -6, -1, -2, -2, 4, 2, 3, -1, -1], [7, -1, 3, 2, -4, 0, -4, 1, -2, 0, -5, 3, 3, -3, 0, -5], [-3, 6, -1, -1, -7, -2, -3, -2, 1, -5, -9, -1, 6, 5, -2, -6], [8, 5, 2, 1, -3, -1, -4, 0, -2, -2, -5, 6, 1, -3, -2, -3], [1, 4, -4, 6, 0, 1, -2, -3, 1, -4, 1, 5, -8, -5, -6, 2], [2, 1, -5, -1, 2, 1, 4, -1, 0, -3, 0, 1, 1, 1, -2, 3], [3, 10, -5, 0, -3, 3, -5, 4, 2, -1, -4, 2, -3, -4, -1, -1], [-5, -3, 4, -1, 5, 1, 2, 0, -1, 2, -2, 0, 6, 2, 4, 3], [5, 3, -4, -3, 5, 0, 2, -3, 2, -3, 1, 3, -2, 2, -3, 1], [-5, -3, 0, -2, -5, 2, 0, 1, 5, -1, -1, -3, 3, 0, 1, -5], [7, 6, -5, 2, -4, -2, -1, 2, -2, -3, 0, 3, -3, 3, -5, 0], [-6, -2, 4, 3, 3, -1, -1, 3, 0, 2, 2, 6, 0, -1, 0, -2], [11, 1, 4, 2, -1, 1, 1, -2, -1, 0, -5, 1, 2, -1, 3, -3], [-6, 5, -9, -1, -9, -3, -7, -2, 4, -5, -7, 0, 2, 2, -2, -4], [4, 3, 0, 0, -7, -3, -7, -7, 2, -3, -3, -3, -5, 1, -5, -1], [0, 4, -6, 3, 0, 0, 4, 2, 0, -1, 4, 5, 3, 0, -4, 0], [-10, 4, -1, 1, -1, -5, -2, -3, -1, 1, 0, -3, 0, 2, 3, 2], [-4, 6, 2, 0, 3, -1, 5, -3, 0, 0, -1, 5, 5, 0, -2, 0], [-14, 1, -4, 4, 5, 0, 3, 3, -2, 4, 8, -2, -5, -4, 3, 7], [-12, 1, 0, 1, 0, 2, 0, -2, -1, 2, 2, -5, 1, 2, 5, 2], [4, -9, -1, -5, 1, 2, -2, 3, 2, 2, -2, 4, 5, 0, 6, -4], [-2, -3, 1, -2, -4, -1, -2, 0, 0, -4, -3, -4, 3, 2, -1, -1], [-5, 5, -10, 4, -3, -2, -1, -3, 1, -6, 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9, 12, 3, -6, 2], [-2, 2, -3, 2, -6, -4, -3, -10, 3, -5, -2, -3, -2, -1, -13, 2], [17, -11, -1, -5, -6, 4, -12, 8, 2, 1, -8, -4, 8, 4, 9, -1], [8, 3, 6, 3, 7, -1, 2, 4, 0, 4, 12, 3, -6, -3, 5, 2], [12, -2, -4, 2, -6, -6, -2, 5, 0, -5, -10, 10, 6, -2, -2, 2], [13, 2, 0, 2, 4, 2, 0, 12, -4, 5, 9, 5, 1, -1, 2, -2], [-3, 11, 6, 3, 5, 3, 3, -9, 2, 4, -6, 3, 4, 4, -1, -3], [3, -1, -1, 2, -4, 1, -1, 2, 0, -1, -3, -6, 0, -5, -3, 1], [18, -1, 1, -4, 6, 0, -2, 6, -1, 6, 5, 3, 3, 1, 13, 2], [8, -2, -4, 5, -1, 0, 1, 3, 2, -3, -2, 6, 8, 4, -6, 4], [12, -14, 4, -3, -7, -1, -8, 6, 0, 2, -3, -11, 3, 3, 3, -2], [12, 0, -4, 3, 4, -1, -1, -2, 2, -1, 6, 8, 0, -2, -3, 0], [7, -12, 18, -6, 0, -5, 2, -2, 5, 7, 5, -10, -8, -4, 6, 1], [15, 0, 0, 2, 2, 2, 0, -1, 0, 1, -1, -6, 4, 0, -2, 5], [-19, 12, -8, 3, -4, -3, -3, -3, 2, -11, -9, 4, 7, 2, -6, -6], [3, -5, 4, 3, 3, 1, -3, 3, -4, -1, 1, 4, -1, 4, -9, 3], [10, 12, -9, 4, -2, 4, -5, -2, 1, -2, 4, -2, -9, -4, -7, 0], [9, -9, 5, 0, -2, -5, -3, 5, -2, -1, -5, 8, 3, 3, 1, -2], [-8, 0, -3, 5, -9, 1, -8, 3, 5, -4, -13, 2, 7, 2, 2, -7], [13, 1, 5, 0, 1, 2, 3, 3, -4, 3, -2, 2, 5, -2, 2, 0], [-11, 11, 6, 1, 4, -3, 3, -10, -1, -1, 0, 8, -3, 1, -4, -4], [1, 9, 10, 5, 5, 0, 2, -4, -2, 4, 6, -9, -3, -3, 3, 6], [3, 6, -8, -4, 0, 2, -3, 4, 2, -3, 0, -3, 2, 0, -6, -1], [28, -9, 11, 0, 3, -1, 2, -2, 2, 7, 7, -7, -2, -2, 5, 1], [0, -4, -1, 4, -1, 0, 2, 0, 2, -2, 3, 4, 10, 3, -6, -2], [14, -2, -3, 3, -4, 3, -7, 7, 0, -3, -7, 10, 3, -4, -1, -11], [-30, -4, -6, -3, -6, -8, -2, 1, 3, 0, 0, -3, 4, 5, 4, 1], [1, 3, -1, 0, 1, 6, -4, 5, 5, -1, 2, 6, -2, -1, 1, -1], [13, 4, -4, 3, 8, 1, 3, 0, -4, 2, 4, 10, 4, -5, -3, -2], [-5, 10, -6, 0, -12, -4, -2, -5, 1, -7, 2, -1, -5, -3, -5, -7], [1, -2, -7, -1, 0, 2, 4, -7, 3, -9, -9, 2, 1, 7, -1, 0], [3, -4, 7, -4, -7, -3, 0, -8, 9, -4, -7, -4, -5, 0, -3, -1], [0, 0, 7, -5, 3, -3, 3, -2, -3, 5, 3, -3, -9, -1, 8, 5], [0, 2, -2, -1, 7, 8, 2, 5, -3, 8, 12, -6, -4, -1, 2, 1], [2, 13, 2, 2, -4, -1, 2, 1, -3, 2, -1, -2, -7, -6, -5, 2], [-4, 3, 1, -7, -3, 0, 2, -2, 5, -5, -8, 2, 8, 5, -3, -11], [7, 6, -1, -6, -11, 3, -6, 3, 4, -9, -15, 5, 16, 2, -3, -13], [1, -4, 6, 0, 5, 1, 0, 5, -6, 3, 7, -3, -1, -1, 7, -4], [-16, -2, -1, -3, 8, 4, 4, 4, 5, 2, -1, -3, -10, -3, 5, 4], [0, -7, -4, 7, 4, -1, 9, 5, -3, 2, 4, 9, -4, -5, -2, 9], [11, 5, 10, -5, 7, 4, -2, 3, -3, 5, 1, -10, 3, 8, 11, -1], [11, -11, -1, -8, 0, 1, 1, 5, 3, 5, -4, -6, 10, 3, 16, 2], [4, -6, 1, -1, -2, 4, 4, -4, 1, -3, -9, 3, 1, 3, 0, 4], [-9, 13, -14, -1, -2, 5, -5, 4, 6, -2, 1, -2, 6, 1, -1, -3], [13, 0, -8, 8, -2, 0, -2, 3, 3, 3, 5, 0, 2, -1, 0, 1], [-16, 7, -7, 9, 6, 0, 12, 10, -4, 1, 11, 11, -3, -7, -1, 7], [-4, 4, -3, 4, -4, -1, 5, -9, 2, -3, -2, 0, -8, 2, -6, 4], [-10, -2, 3, 2, 6, -4, 0, -5, -1, 6, 10, -6, -6, -4, -7, 3], [-3, 1, 6, -2, 2, 4, 4, -5, 0, -3, 0, 6, 1, 2, -7, -1], [-4, -1, -6, -2, -3, 1, -2, 5, -1, 2, -3, -7, -6, 2, -2, 3], [4, -8, 2, -4, -4, -4, 5, 0, -2, 0, -5, 8, -1, -9, 1, -1], [13, 5, -5, -1, 1, -1, -3, -8, 0, -8, -5, 3, 0, 16, -11, -2], [9, -15, -2, -6, -10, 0, 0, 2, 2, -1, -12, -5, 10, 5, -1, -5], [11, -6, 6, 4, 3, 6, 2, 8, -5, 5, -5, 1, 13, 1, 4, 3], [-6, 1, -1, 7, 0, -4, -2, 1, 3, -1, -5, 3, 3, -4, 1, 8], [19, 7, -5, 3, -7, 9, -6, 14, -2, 3, -1, -8, 1, -3, 11, -1], [8, -12, 1, -3, -1, -1, -4, 11, -3, 2, -2, 6, 0, 3, 0, 1], [-1, 3, 1, -2, 1, -4, 7, -7, -1, 0, 4, 5, -3, 2, 1, 6], [-4, 0, -4, -3, 5, -2, 4, 11, -3, 3, -2, 10, 6, 3, 8, 1], [11, -3, 10, 2, -3, -3, -2, 2, 2, 4, 5, -7, -2, -8, 8, -4], [-4, 3, -4, -3, -17, -8, -1, -9, 5, -11, -7, -5, -4, 4, -8, -7], [11, -2, 1, 0, -1, -7, 6, 2, 2, 0, 5, -4, -2, -5, 2, 8], [-3, -1, 0, 5, -2, -2, -2, 0, 0, -3, -3, 9, 3, 5, -10, -4], [-32, 2, -15, 2, -8, -6, 1, -8, 7, -9, -7, 3, -3, 2, -9, 3], [7, 3, -7, 12, 0, -3, -7, -4, -3, -5, 4, 5, -11, -4, -11, 6], [-12, -7, -3, -2, -1, -2, 0, -7, 5, -3, 2, 8, -7, -2, -6, 2], [-8, 19, -6, 3, -3, -5, -3, -4, 1, -7, -9, 10, 9, 2, -6, -6], [1, -5, 6, 3, -3, -8, -3, -7, 4, 0, 2, -1, -2, 1, 5, 5], [9, 4, 5, 1, -3, -1, -2, -4, -3, 2, 6, 5, -2, 1, -10, -8], [-8, 7, -8, 5, -1, -5, -2, -13, 6, -6, -2, 4, 0, 4, -5, 2], [1, 6, -6, 0, -4, -1, 0, -2, -2, 0, -3, -1, 9, 4, 6, 0], [-6, -8, -6, 2, -3, -3, 0, -3, 4, 0, 9, -13, -14, 0, -6, 8], [23, -4, -7, 5, -2, -3, -6, -1, 0, -5, -7, 3, 6, 6, -15, -6], [-15, -10, 4, -1, 10, 7, 7, -1, 3, 7, -2, -6, 13, 0, 7, -1], [24, 1, 7, -5, -3, 6, -6, 3, -2, 2, 0, -10, 5, 0, 4, -8], [-10, 0, 3, -3, 8, 8, 5, -2, 2, 9, 4, -2, -1, 0, 10, 0], [-16, -1, -10, 3, 4, -6, 3, 4, -1, 2, 5, -1, -4, -3, -4, 7], [5, -1, -5, -7, -7, 6, -1, 10, 3, -4, -13, 6, 8, 3, 5, -10], [-5, 5, -1, 2, 2, 1, 1, 1, -8, 5, 4, -3, -3, -1, 0, 0], [-5, 0, 6, -4, -9, 3, -2, 2, 3, -1, -2, -12, -3, -6, 6, -6], [25, 5, -4, 3, 2, 5, 4, -1, -3, -8, 2, 16, 3, -3, -10, 0], [-6, -11, -6, -7, -3, 1, -5, 10, -2, 5, -4, -10, 5, -2, 16, 4], [16, -1, -9, -4, 4, 1, 2, 1, -6, 1, 5, -2, -5, 6, 0, -1], [21, 10, 2, 6, 3, -1, -4, -1, 1, -10, -1, 11, 0, -2, -15, -2], [-22, -4, 5, 1, 5, 1, 6, 4, -7, 9, 14, -1, -9, -6, 8, 1], [3, 2, -9, 3, -3, -3, -3, -3, 5, -5, 1, 2, -11, -1, 1, 8], [-13, -2, -2, -7, -15, -8, -7, -4, 3, -5, -6, 0, -4, 0, -1, -6], [-6, 6, 2, 0, -8, -6, -2, -7, -2, -7, -9, -2, 6, 5, 1, -1], [17, -2, 5, -1, 4, 2, -1, -4, -1, 5, 5, -6, -3, 7, 4, 1], [-12, -6, 13, -5, -3, -7, -3, 3, -3, 4, -6, 0, 8, 3, 4, -2], [-6, 14, -17, 1, -7, 4, 7, -6, 4, -7, 0, -3, 1, -4, 1, -3], [-8, -8, 3, 0, -1, -1, 1, -1, 1, 5, 3, -8, 5, 5, 2, 2], [-5, 8, -3, -2, 0, 0, 8, -1, -5, -1, 0, 4, 6, 3, -1, -1], [3, 19, -7, -1, -7, 1, -6, -2, 0, -12, -4, 5, 4, 7, -15, -11], [0, 1, -6, 2, 5, 4, 7, -5, 1, -2, 5, 0, -10, -4, -8, 5], [-4, 3, 11, -1, -1, 2, -5, -4, 3, 1, -2, -9, 2, -4, 4, -6], [-22, 4, -5, -4, 0, 8, 7, -1, 6, 0, -2, 2, 5, -1, 6, -9], [-1, -7, 5, 3, 3, -3, 3, -9, 1, 5, 13, -5, -7, 5, -1, 2], [6, 5, 5, -1, -2, -2, 0, 11, 0, 1, -4, 5, -4, -6, 0, 3], [-14, -5, -6, 0, 1, -3, -2, 1, 1, 7, 9, -9, -6, 2, 10, 10], [9, -3, -5, 0, -8, -7, -10, 5, 4, -4, -14, 2, 6, 4, 1, 0], [-11, -2, -7, 0, 4, 2, 6, -10, 1, -2, -2, 1, 1, 4, -1, 1], [-1, -12, 7, -5, 2, -10, -3, -3, -4, -3, -1, 3, -4, 3, -8, 2], [4, 7, -8, 3, -6, -6, -9, 3, -1, 1, -1, 0, -10, -4, -2, 8], [15, 5, 3, 0, 2, 0, -6, 3, 2, -5, 3, -6, 1, 0, -1, -7], [-5, -5, 1, -5, -4, 6, 4, 5, 8, 6, -1, -3, 8, -6, 14, -7], [20, -8, 6, -8, 1, 10, 2, 3, 2, 5, 0, -8, 5, -1, 5, -8], [15, 0, -11, 6, -3, 1, 2, -4, -3, -1, 5, -5, -11, -3, 0, 1], [-3, -5, 13, -3, 2, 3, -4, 5, 2, 6, -5, -2, 3, 2, 9, 2], [0, 0, -9, -1, -4, 2, -9, -5, 1, -4, -9, 1, -6, 3, -2, 3], [-12, -14, 8, -7, -6, -4, 0, 8, -2, 4, -8, -5, -1, 0, 19, 1], [-4, 4, -8, 6, 5, 4, -1, 0, 5, -1, 6, 6, -12, -2, 0, 8], [27, -17, 7, -5, -4, -4, 0, -2, 9, -4, -6, -7, 4, 4, 2, -2], [18, -2, 0, -3, -13, -5, 1, 1, 3, -6, -7, 5, -2, -2, -7, -7], [-6, -14, 8, -7, -6, 3, 10, 1, -4, 5, 3, -11, 5, 2, 10, -4], [7, 4, 1, -1, -4, 1, -1, 3, -1, 3, 2, -8, -5, -1, 15, 4], [-12, 18, -4, -1, 2, 6, -3, 4, -2, 0, -4, -1, 10, 0, 11, -3], [-5, -1, 0, -3, 2, -2, 0, 4, -4, 3, -2, -2, 8, -3, -8, 0], [3, 6, -4, 1, 5, 4, 1, 10, -2, 8, 12, -2, 4, -1, 8, -5], [-9, -3, 0, 3, -4, -2, -8, -2, 7, 6, 2, 4, -2, -8, 5, -4], [-12, -10, -6, -2, -7, -6, -8, -7, 1, -11, -10, 10, 0, 8, -8, -9], [2, -8, 5, -6, -2, -1, 0, 4, -5, 9, -3, -6, 0, -2, 13, 7], [1, 7, -10, 1, 0, 5, -2, 2, 2, -1, -2, 6, 4, 0, 9, 2], [15, -14, 7, -3, 1, 4, -1, 19, -5, 8, 2, -4, -1, -9, 11, -2], [-17, 9, -2, -6, -2, 1, -3, -6, 7, -2, -4, -14, 6, 8, 9, 3], [8, 1, -10, 4, 3, 3, 0, 5, -4, -2, 7, 9, 3, -1, -10, -4], [2, 3, -5, 3, -2, -3, -11, 6, -3, 0, 0, -3, -7, -4, -3, 4], [7, 10, 9, 8, 11, 8, 8, -7, -4, 7, 11, -3, -2, -1, -1, 1], [6, 15, -4, 2, 0, 3, -7, 5, 3, -2, -6, 7, 10, 2, 2, -8], [3, -1, -2, 4, 12, 2, 0, -2, -2, -1, 9, 4, -7, 3, -12, 7], [20, 9, -9, 6, -4, 3, -5, -8, 5, -14, -16, 7, -1, 1, -4, -2], [8, -3, -6, 5, 1, -3, -3, -10, 1, -10, -6, -1, -3, 8, -17, 1], [5, -14, 6, -10, -5, 1, 1, 1, 1, 3, 0, -6, -5, -8, 6, -1], [-11, -2, 0, 0, 3, 2, 4, -10, -2, -2, 4, -5, -3, 7, -7, 6], [-14, 11, -17, 4, -11, 0, -6, -7, 3, -14, -10, 8, 1, 4, -8, -7], [-4, 12, 9, -1, 4, 0, 9, -5, -4, 9, 11, -12, -15, -9, -2, 1], [8, 1, 0, -3, 10, 2, 6, -5, -4, -4, -1, 12, 5, 9, -5, -2], [40, 9, -12, 9, -13, -3, -9, 0, 6, -13, -5, 7, -7, -8, -18, -7], [-5, 18, -15, 6, 1, 1, -1, -5, -1, -6, 1, 1, 9, 7, -9, -3], [-17, 3, 12, -1, -3, 4, -5, 11, 1, 0, -8, 2, 9, -5, 7, -5], [-31, 11, -9, 1, -14, 1, 0, -4, 7, -8, -9, 6, 5, -1, -9, -13], [1, -6, -9, 3, 3, 3, -1, 4, 3, 0, -1, -5, -8, -2, -1, 10], [2, 8, -4, 3, -13, 1, -5, -5, 4, -11, -15, 13, 14, 7, -12, -11], [9, 2, -6, -5, -6, 2, -3, 10, 2, -4, -16, 12, 17, 6, -4, -2], [-15, 1, 1, 1, 5, -2, -2, 3, -4, 3, 1, 5, 3, -7, 3, -2], [-1, -21, 9, -3, 11, 4, 10, -2, 0, 17, 17, -17, -6, -9, 21, 9], [8, 10, -8, 1, 3, -8, 2, 6, -6, 0, 1, 5, 3, 6, -4, 7], [-10, -2, -8, -1, -5, 0, -2, 7, 6, -3, -7, -5, 1, -2, 3, 0], [10, -1, 3, -12, -4, 6, -8, 7, 2, -2, -13, 1, 8, 2, 4, -11], [5, 3, -1, 6, 7, 5, -10, 5, -1, 5, 4, -1, -14, -7, 4, 3], [24, -5, -4, 6, -3, -7, -7, 3, -6, -2, -5, 5, -2, -3, -2, 5], [5, -1, -3, -5, -4, -4, -5, -4, 2, -13, -2, 13, 7, 7, -7, -8], [4, 2, 1, 3, 7, -3, 1, -3, -8, -1, 0, 12, -1, 0, -1, 6], [-10, 7, -15, 3, -11, -1, 1, -10, 5, -8, -8, -1, 2, 5, -5, 0], [-1, -23, -2, -2, 10, -1, 9, -4, 4, 5, 12, -2, -9, 3, -8, 14], [-21, 11, -9, 1, -6, -1, -3, -5, 0, -7, -16, 10, 9, 5, -7, -11], [0, 10, 0, -2, -2, -8, 0, -11, 1, -6, -6, 4, 0, 3, -9, 6], [-13, -2, -5, 1, 1, 7, 4, 4, -2, 7, 7, -15, 1, 5, 14, 8], [-15, 12, 1, 4, 6, 0, 7, -3, -7, 1, 11, 6, 2, 4, -3, 0], [3, 5, -1, 1, -2, -5, -5, -3, -4, 0, 4, 0, -5, -3, -7, -4], [7, 5, -4, -2, 1, 2, 4, 2, -2, 6, 15, -2, 1, 0, 4, -8], [-10, -5, -2, -3, -9, -2, -1, 0, 4, -4, -9, 1, 9, 3, -9, -9], [7, -13, -5, -3, 0, 5, -5, 8, 2, 5, -3, -3, 14, 5, 9, 0], [27, -6, 10, 2, 10, 3, 2, 3, 3, 5, 14, -8, -9, -7, 7, 13], [-16, 5, -5, 8, -4, 6, -6, 6, 2, 2, 0, -5, -3, -3, 1, 4], [7, 9, 0, 2, 2, 2, 5, -7, 5, 1, 9, 9, -8, -3, -9, -6], [-3, -5, 11, 5, 4, -8, 6, -12, -5, 6, -3, 2, 1, 2, -9, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6045_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6045_2_a_bg();".
// 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_6045_2_a_bg(:prec:=16)
    chi := MakeCharacter_6045_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_6045_2_a_bg();".
// 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_6045_2_a_bg( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6045_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-428, -2178, -780, 8572, 6027, -11579, -8358, 7583, 5124, -2678, -1657, 517, 294, -51, -27, 2, 1]>,<7,R![-16000, 56320, 392664, 503244, -96326, -496367, -124280, 179258, 70634, -31113, -14583, 2769, 1406, -121, -62, 2, 1]>,<11,R![4757120, -2429888, -21560456, 2852420, 27216926, 4663297, -8947057, -1662483, 1375871, 202974, -112330, -11437, 4971, 302, -112, -3, 1]>],Snew);
    return Vf;
end function;