
// Make newform 6672.2.a.bs in Magma, downloaded from the LMFDB on 16 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6672_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_6672_2_a_bs();".
// 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_6672_2_a_bs();".
// 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 := [-4912, 14748, -5172, -15919, 7482, 7294, -2549, -1508, 397, 142, -31, -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], [5628424, -22906682, 22135189, 10043776, -18324226, -849263, 4773986, -267075, -517498, 56733, 20325, -3017], [-5796368, 22858698, -22111197, -10043776, 18324226, 849263, -4773986, 267075, 517498, -56733, -20325, 3017], [27868432, -92360156, 69534596, 46776931, -59019221, -6356469, 15372130, -466058, -1661107, 163257, 64914, -9313], [-11657184, 39520206, -30406939, -19947233, 25623409, 2682904, -6653644, 207515, 717295, -70654, -27977, 4014], [-43629184, 156722800, -131840054, -76978839, 111065777, 9645727, -29095690, 1095852, 3169279, -320223, -125024, 18091], [-80234704, 284227620, -232704064, -142433629, 196790895, 18908655, -51630002, 1712406, 5633241, -556459, -222678, 31999], [96039728, -332685868, 265197496, 166977135, -224011145, -22050013, 58560418, -1955758, -6362055, 632169, 250246, -36033], [89825264, -322623340, 272798992, 155030133, -227551587, -18057231, 59051038, -2504010, -6366725, 660819, 248378, -36227], [58053024, -199287028, 157157734, 100902245, -133593807, -13734869, 35041390, -1079848, -3818821, 373829, 150712, -21601], [-139131968, 496376032, -413153534, -244653463, 348742137, 30918503, -91416754, 3389612, 9960871, -1004567, -392992, 56843]];
    Rf_basisdens := [1, 1, 23992, 23992, 47984, 11996, 47984, 47984, 47984, 47984, 23992, 47984];
    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_6672_a();"
function MakeCharacter_6672_a()
    N := 6672;
    order := 1;
    char_gens := [5839, 1669, 2225, 4033];
    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_6672_a_Hecke(Kf)
    return MakeCharacter_6672_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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0], [1, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0], [2, -2, -1, 0, -1, -1, 1, 0, 0, 0, 1, -1], [2, -1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0], [0, 1, 0, 0, 0, 0, 1, 1, -1, 0, 0, 0], [1, 2, 0, 1, 1, 1, 0, 1, -1, 0, 0, 0], [2, 0, 0, 0, 1, 0, -2, 1, 0, 1, 0, 0], [0, 0, -1, 0, 0, 0, -2, 0, 0, 0, 0, 0], [3, 1, 0, -1, 2, 1, 0, 0, 0, 0, 0, 0], [2, 1, 0, 1, 0, 0, -1, 2, -1, 1, 1, -1], [0, -2, -1, 1, -1, -1, 1, 0, 1, 0, 0, -1], [4, 2, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0], [0, 1, 0, 1, 1, 1, -3, 1, -1, 0, 0, 0], [2, -1, -1, 1, 0, -1, 1, 0, -1, 1, 1, -1], [2, -2, -1, -1, -1, -1, 0, 1, 0, 1, 1, -1], [4, 1, 0, 0, 1, 1, -1, -1, 1, 0, -1, 0], [4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1], [-1, -3, 0, -1, -1, 0, 2, -2, 2, 0, -1, 0], [0, -1, 0, 0, 0, -1, -2, 0, 1, 1, -1, 0], [1, 1, -1, -1, -1, -1, -2, 1, 0, 1, 0, 1], [2, 0, 0, 1, 0, -1, 0, 0, 2, 0, 0, 0], [1, 2, 2, 0, 1, 2, 0, 0, 0, -1, 0, 0], [0, 3, 1, 1, 1, 1, 1, 1, -1, 0, 1, 1], [3, 1, 0, 2, 0, -1, -1, -1, -1, -1, -1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 3, 1, 2, 1, 2, -1, 1, -1, 0, 0, 1], [-3, 3, 1, 1, 1, -1, -2, 1, -2, 1, 0, 1], [5, 1, -1, 1, -1, 0, 0, -1, 0, -1, -1, -1], [-2, -1, -2, -1, -1, 0, -1, 1, -1, 0, 1, -2], [-3, -1, -1, -1, 1, -1, -1, -1, 2, -1, -2, 1], [0, -2, 0, 1, -2, 0, -1, 1, 1, 1, -1, 0], [-2, -2, -1, 2, -2, -1, 2, 0, 0, 0, 1, 0], [2, 3, 1, -1, 1, 1, 1, 0, 1, -1, -1, 1], [-1, -2, 0, 1, 0, -2, 0, 0, 3, 1, -1, 0], [4, -3, -2, -3, -2, 0, 1, -1, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0, -2, 0, 2, 0, -2, 1], [-7, 0, -1, 1, -1, -2, -1, 2, -3, 1, 1, -1], [2, 2, -2, 2, 1, 2, -2, 0, -2, 0, 0, 0], [-1, 1, 1, 1, 2, 2, 2, -1, 0, -1, 1, 1], [0, 1, 1, 0, -1, -1, 3, 1, -1, 0, -1, 0], [3, 4, 1, 1, 2, -1, 1, 2, 1, 1, 0, 1], [0, -1, 0, 0, 0, 3, 1, -2, 1, -1, 1, 1], [-3, -4, -1, -1, -3, 0, 3, -1, 1, -2, -1, -1], [10, 2, 1, 0, 1, 2, 0, -1, -2, -1, 2, 1], [2, 2, 0, 1, -2, -2, 0, 4, -2, 0, 1, -2], [0, 2, 2, 3, 0, 0, 2, 0, 2, 0, -1, 2], [-1, 1, -1, 0, 2, 2, -4, 0, 0, 0, 0, 1], [2, -2, -1, -2, -1, -2, 2, 1, 2, 1, 2, -1], [-3, 1, 0, 3, -1, 0, 0, 0, 0, 0, 1, 2], [7, 2, -1, 1, 2, 1, -1, -2, -1, -1, 0, -1], [0, -2, -2, 0, -1, -1, 0, -1, 0, -1, 1, -2], [2, -1, 1, 0, -1, 1, 1, -3, 1, 0, 1, 0], [-2, 3, 2, 4, -1, 0, 1, -1, -1, 0, 0, 0], [0, -1, 0, 0, 1, -1, 0, -1, 3, -2, -1, -1], [2, -1, 1, 0, 2, 1, 1, -2, -1, -1, 3, -1], [8, -2, 1, -2, 2, 3, 1, -1, 0, -1, 1, 0], [-2, 4, 1, 0, 4, 2, 0, -2, 2, 0, 0, 1], [0, 7, 2, 0, 1, 2, 1, -1, -1, 0, 0, 2], [4, 3, 1, 2, 0, 1, 1, 0, -1, 1, -1, 1], [2, 7, 1, -3, 1, 2, -1, 3, -1, 0, 1, 1], [2, -5, -2, -3, -4, -3, -1, -1, 1, 0, -1, -2], [-6, 1, -2, 0, -3, -2, -3, 3, -3, 2, 0, -2], [-3, 0, 2, -2, 0, 1, 5, 0, 1, -3, 0, 0], [1, 2, 3, 2, 0, 0, 2, -1, 0, 0, -2, 1], [1, 2, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1], [-2, 8, 2, 2, 3, 2, -2, 4, -2, 2, 0, 0], [-4, -1, -2, -2, -1, -2, -1, -1, 1, 0, 0, 2], [-6, -2, -1, 2, -3, -2, 4, -1, 2, -1, -2, -1], [3, 4, 1, 1, 2, 3, 1, 2, -1, 1, 2, 1], [-8, 0, -1, -2, -1, 0, -2, 1, 2, -1, 0, 0], [11, -1, 2, 0, 0, 0, 3, 0, -1, 0, 2, -1], [0, -2, -2, 0, -2, -3, 0, 0, 0, -2, -1, 0], [-3, 0, -2, -2, -1, -2, 0, 4, -2, -1, 2, -2], [-4, -1, -1, 2, 0, -1, 3, 0, 1, -1, -3, 1], [5, 1, -2, 0, -2, -2, 1, 1, -1, -1, 0, -2], [-7, 0, -1, -3, 0, -1, 3, -2, 1, 1, 0, -1], [10, -1, -2, -4, -2, -2, -1, 0, 1, 1, 0, -1], [-2, 4, 2, 4, 3, 1, 0, 5, -2, 1, -1, 0], [4, -1, 2, -2, -1, 0, -3, -1, 1, 2, 0, 0], [-5, 2, 0, -2, 1, -3, 1, 1, -1, 0, -1, 0], [2, -3, -1, 4, -4, -2, -1, 0, -3, 3, 0, -1], [12, -1, 2, 4, 1, 0, 1, -1, -1, 0, 0, 0], [-2, -3, 0, 2, -3, -1, -1, -1, 3, 0, -1, 0], [-5, 1, -1, -3, -2, 2, 4, -1, 0, -1, 3, -1], [4, 2, 1, 0, 2, 1, -2, 2, 0, 0, -1, 2], [2, -2, 3, 0, 0, 0, 6, 1, 2, -1, -2, 1], [-2, 0, 2, -3, 2, 5, 3, -3, 1, -1, 0, 2], [0, 0, 1, 2, 0, 4, -2, -2, -2, -2, -2, 1], [-4, 4, 0, -4, 2, 2, -2, 2, 0, 0, 0, -2], [6, -3, -2, -2, 0, -2, 6, -2, 1, -1, 2, -2], [-2, 2, 2, 2, 2, 1, 3, -3, 0, -1, -1, 4], [2, -3, 0, -2, -2, 0, -1, 1, 3, 2, 2, 2], [-3, 0, 1, -1, 0, 1, 1, 4, 1, 1, -2, 1], [5, 0, 1, 2, 1, 2, 2, 0, -4, -1, 2, 1], [-10, -1, 1, -3, 2, 1, -1, 0, 3, -1, 3, 1], [2, 3, 0, -6, -1, 0, -1, -1, 1, -2, 2, 2], [4, 0, 2, 4, 0, -1, 0, -4, 2, 2, -1, 2], [16, -2, 2, -2, 0, 1, 1, 2, 0, 0, 1, -2], [-2, -4, -1, -1, 1, 1, -2, 1, -2, -1, 3, -1], [-4, 0, 2, -2, 0, -2, -2, 0, 2, 0, -2, 2], [-1, 1, -1, 4, -1, -2, 1, 3, -1, -1, -2, 0], [8, -3, -1, -1, 2, 1, -1, 0, -5, 1, 1, -1], [2, 0, 0, 1, 2, 0, -2, -2, 0, -2, 1, -2], [-14, -2, -3, 0, -2, -1, -6, 2, -2, 2, -1, -3], [-4, -4, -1, -2, 1, 1, -3, -3, 2, 1, -3, 2], [-1, -4, -1, -1, -2, 1, 1, -2, 1, -1, 0, 1], [-6, 0, 1, -2, -2, -1, -2, 0, 0, 2, 1, 3], [12, -3, 2, 2, 2, 0, -3, -1, -1, -2, 0, 0], [-4, -1, 0, 5, 0, 0, -2, 5, -2, 2, 1, 0], [-6, 3, 1, 4, 3, 2, -1, -1, -1, 2, 0, -1], [6, 4, 3, 4, 3, 3, -4, -1, 0, 3, -5, 3], [1, 4, 1, 1, 0, 0, -1, -2, -1, -1, 1, -1], [-2, -6, 0, -3, 1, 1, 0, -1, 0, -1, 0, 1], [-9, 5, 3, 1, 1, 3, 2, -3, -2, -1, 2, 1], [0, -6, -1, -3, -2, -3, 2, 0, 4, 4, 1, -1], [14, -3, 1, 2, -1, -2, 1, 0, 1, -1, -2, -1], [0, -5, 0, -1, -3, 1, 2, -2, 2, -7, -2, 0], [-3, -3, -2, -1, -2, 0, 2, 0, -2, -2, 3, -4], [-11, 2, -1, -4, 3, 3, -3, 3, -1, 0, 2, 1], [3, 1, -1, -1, -1, -1, -2, -1, 0, -1, -2, -3], [2, 3, 1, 0, 2, -1, 4, 1, 3, 0, -3, 1], [-6, -3, -3, 3, -2, -2, 3, 2, -3, 1, 4, -1], [-4, 4, 1, 0, -3, 0, -2, 1, -4, -1, 0, -1], [13, 2, 3, 5, -1, 2, 1, -2, -1, -1, -1, 3], [-5, 2, 2, 1, 1, 0, 3, -3, 3, 0, -1, -2], [6, -2, 1, 3, 1, -3, 0, 1, 0, 1, -2, -3], [6, 5, -1, 0, 4, 2, -1, 2, -3, -1, 0, -1], [9, 0, 4, 4, 1, 2, 3, -5, 3, -2, -3, 2], [8, -8, 0, -2, -3, 0, -2, -2, 2, -2, -2, 0], [2, 0, 1, 2, -2, -2, 0, 2, 2, 0, 2, 3], [-2, -5, 1, -2, -3, -1, 5, -1, 3, 2, 1, -1], [9, -6, 0, 6, -2, -1, 2, 1, 0, -2, 1, 3], [-10, 1, -1, -2, 2, 4, 1, -4, 1, 1, -2, 1], [-1, 4, 2, 1, 0, 0, 1, 3, 1, -2, -1, 4], [12, -7, -3, 0, -1, 4, 1, -5, 1, -2, 4, 0], [5, 2, 1, 1, 2, -4, -3, 2, 1, 1, -5, -1], [-4, 4, 4, 1, 5, 2, -3, -2, 1, -2, -1, 0], [-9, 6, 2, 1, 3, 0, -1, 7, -1, 2, 1, 2], [-10, -1, 1, 4, 0, 0, -1, -4, 1, -3, -4, 2], [11, -3, 3, 0, 3, 1, -5, 0, -1, 4, -3, 1], [-8, 2, 2, 1, 2, 1, 1, -3, -1, -1, 0, 2], [1, 2, 1, -1, 0, 2, -1, -2, 1, 1, 5, 3], [0, -3, -1, -4, -3, -3, 2, -1, 1, 2, -1, -3], [10, 1, 0, 1, -1, 3, -5, 1, -3, 2, 5, 2], [-6, 6, 0, -2, 2, 1, -4, -6, 2, 2, -1, 2], [2, -6, -4, 2, -2, -4, -6, 2, 2, 2, 0, -2], [-5, 2, -1, 1, -2, 3, 2, -2, 1, -3, 0, -1], [-17, 12, 5, 1, 2, 3, 1, 0, 1, 1, 0, 1], [-2, -6, -1, -6, 0, -1, 6, -6, 2, -2, 3, -1], [-6, 1, -3, -4, 0, -2, 3, 2, -1, -1, 4, -5], [5, -2, 0, -3, -2, 1, -2, -6, -1, -1, 2, 0], [7, 7, 2, 0, 1, 4, 5, -4, 1, -4, 0, 4], [-6, -3, 0, -4, -2, -1, 1, 2, -1, 5, 3, -2], [3, -4, 2, -3, 0, 0, 7, -3, -1, -4, 1, 0], [2, 2, 3, 1, -2, 1, 4, -2, 2, -4, -1, -1], [-4, 3, -3, 3, -2, 0, -1, 6, -3, -1, 0, -3], [-6, 3, 2, 2, -3, -1, 5, 1, -1, 0, 1, 0], [1, -3, 3, -1, 2, -2, 4, -3, 6, -1, -3, 3], [-12, 3, 0, 4, -4, -4, -5, -1, 1, 2, -2, 2], [16, 7, 3, 3, 0, 4, 1, 1, -1, 0, 1, 0], [2, -1, -3, 2, 1, 0, 1, 1, 1, 0, -6, -1], [-8, 3, -1, -1, 1, -2, -3, 1, 1, 4, 3, -1], [7, -2, -1, 3, -4, -3, -3, 0, -1, 5, 2, -1], [-14, -2, -4, 4, -4, -4, 0, 4, 0, -4, -2, -2], [10, 1, 1, -2, 2, 1, -5, 0, -3, 1, -1, 1], [-18, 0, -2, 2, -4, -5, 0, 0, 2, 2, 1, 2], [-6, 5, 0, -2, 0, 1, -3, 2, -1, 3, 3, 3], [6, -2, -1, -4, 1, 6, 5, 1, -2, -3, 4, -2], [10, 3, 0, -4, 2, 2, 1, -2, 1, 3, -2, 4], [-3, -8, -3, 1, -4, -3, -1, 4, -1, -1, -4, -3], [4, 7, 4, -4, 7, 7, -2, 0, 1, -1, 3, 2], [1, 2, 1, -1, 3, 2, 5, -2, -1, -1, 3, 1], [-4, 0, -2, -2, -2, -1, 1, 2, -2, 2, -1, 2], [-12, -1, -1, -3, -2, -1, 3, -2, 5, -1, 3, 1], [-7, 3, -2, 2, -2, 0, -5, 1, -3, 5, 0, 0], [-16, 6, 0, 2, 2, 0, 4, 0, -4, 2, 0, 0], [-3, 4, -1, -1, 3, 0, 0, -3, -3, 2, -3, -1], [7, 0, -3, -4, 3, 2, 0, 3, 2, 2, 0, -3], [-4, 7, 2, 3, 1, 3, -1, 5, -3, 6, 4, 0], [-6, -2, -2, -4, -2, 1, 0, -4, -2, -2, -3, 0], [0, -4, 2, 4, 0, -4, 2, 2, 4, 2, -2, -2], [11, -7, 5, 0, 4, 5, 1, -4, 1, 2, -1, 1], [12, -5, 3, 2, 0, -4, 3, -6, 1, 1, -4, 3], [14, -5, -1, 4, -3, 0, 7, 0, 1, -3, -2, 1], [13, -4, -4, -3, -3, 0, 5, -1, -3, -2, 1, -4], [-6, 4, -1, 6, -3, -1, -8, 1, -4, 1, 1, -1], [-14, 7, 0, 7, -5, -6, -1, 3, -1, 4, -3, 0], [1, 8, 3, -2, 1, 5, 0, 0, 2, -1, -3, 2], [9, 0, 2, 3, 1, -2, 3, -3, -3, 0, 5, -2], [-6, 9, 1, 2, 5, 5, -3, 5, -3, 6, 3, -1], [-12, -3, 1, 0, -4, -4, 1, 4, 1, 1, 0, 3], [5, 0, 0, 0, -3, 1, 1, 1, -1, -4, 4, 0], [9, -1, 2, 1, 2, -1, -2, 2, 4, 4, -2, -2], [-14, 2, 2, -3, 3, -1, -4, 0, 2, 2, 1, 2], [10, -1, -1, -4, -3, -1, -6, 0, 1, 1, -1, 0], [-18, -1, 1, 2, 4, -1, 1, 0, -1, 1, 1, 3], [8, -3, 0, -4, 0, 0, -1, 6, 3, 3, 2, -2], [-5, 6, 0, 6, 1, 3, 0, 2, -2, 3, 3, 2], [-10, 8, -1, 0, 0, -3, 2, 4, -2, -2, -3, 2], [-16, 3, 2, -2, 0, 0, 2, -2, 3, -3, -2, 2], [4, -7, -3, -2, 1, 0, -3, 0, 3, 5, 2, -3], [10, 5, 4, 9, 3, 0, -1, -2, -1, -1, -3, 3], [-4, 5, 0, 4, 0, -6, -1, 4, 1, -3, -4, 2], [-12, -10, -5, -1, -5, -5, 2, -3, 2, -3, -1, -3], [26, 1, 1, 2, 1, -4, 1, 1, 1, -2, 0, 2], [-30, -1, -2, -4, -1, -4, 1, -1, 5, -2, -4, -2], [-10, 0, -1, -2, 1, 0, 4, -1, 4, -3, -2, 5], [-1, 10, 0, 1, 4, 2, -5, 3, -3, 0, 1, 0], [-10, 2, 2, -2, 1, 3, 8, -3, 0, -1, 3, 6], [-5, -13, -6, -5, -2, -3, -3, -2, 4, -2, -2, -4], [-2, 9, 6, -1, 1, 2, 6, 4, 0, -1, -1, 4], [-5, -2, -2, -7, 5, 3, -3, -1, 1, 2, -2, 0], [-12, 3, 1, -1, 0, 1, 1, 6, 1, -3, 5, -1], [-8, 2, -1, 2, 0, -3, -2, 0, -2, 0, -1, -2], [9, -3, 5, 5, 1, 3, 2, 1, 0, 3, 4, 1], [-4, 0, 0, 9, -3, -2, 2, -3, 0, 1, 3, 0], [-9, -3, 0, -7, 2, 4, 2, -2, -2, 2, 5, -2], [-6, -7, -1, 0, 2, 0, 1, -6, 3, 1, 0, 4], [-3, 2, 5, -2, 5, 4, 1, -1, 1, -4, 0, 1], [-12, 12, 5, 2, 7, 3, 0, 5, 0, 3, -5, 3], [0, 5, -2, 0, 4, 2, -3, -2, -5, -1, 0, 3], [8, -1, -2, -1, 0, -2, 1, -1, -3, 2, -1, -2], [-12, -2, 2, -9, -2, 0, 2, 0, -2, 2, 1, 0], [5, -1, -3, -2, 1, -5, -1, 0, -1, -4, -3, -1], [-4, 3, 8, 3, -3, -1, 3, 1, 1, 4, 2, 2], [18, -2, 1, 4, 0, -8, -2, 1, 0, -1, -4, -3], [-4, -9, 0, -6, 1, 3, -5, -9, 3, -4, -3, 4], [16, -1, 1, 3, 1, 3, -7, 0, -3, 1, -2, -1], [-10, 12, 4, -6, 5, 4, -6, -1, -2, -1, -4, 4], [2, -7, 2, 4, 2, -2, -3, 2, -1, 1, 0, -1], [-21, 0, 1, -2, -2, -2, -3, 6, -3, 7, -1, -1], [-6, -3, -1, -4, 3, -3, 1, 3, 3, 0, -3, -4], [8, -1, 0, -2, -2, 0, -1, -5, 3, 4, 2, 4], [-7, 10, 2, 7, 4, 2, 1, 7, -3, 2, 5, -2], [-7, 6, 1, -4, 1, 0, 2, -3, 4, -4, -2, -1], [-8, 15, -2, 7, 1, 2, 1, -1, -3, -2, 2, 0], [-18, 8, -3, 1, -2, 0, 1, 1, -1, 1, 3, 1], [0, -8, -3, -1, -1, 5, 4, -5, 0, -3, 4, -3], [-8, -1, 1, 0, 0, 5, -6, 2, -1, -3, 1, 1], [-6, 0, -1, -2, 6, 2, 2, -3, 4, 1, 2, -3], [-11, -6, -4, 5, 0, -2, -5, -2, 3, 1, 5, 0], [3, 1, 2, -5, 0, 1, 6, 0, -4, -4, -2, -2], [-2, -5, -4, -6, -6, -5, -1, -2, 1, 7, -1, -2], [-13, -3, 3, 0, 6, -3, -4, 4, 2, 4, -6, 1], [-6, 5, -1, 3, 2, -1, -3, 7, -3, 2, 3, -4], [8, 4, 1, 0, 3, 4, 4, -7, 2, -3, 0, 1], [8, -2, 2, -2, -2, 2, -2, 4, -2, 6, 6, -2], [-1, -8, 1, 3, -2, 2, 7, -10, 5, 1, -1, 3], [2, 15, -1, -4, 3, 2, -5, 2, 1, -1, -4, 1], [2, 3, -4, 6, -2, 0, -11, 6, -9, 3, 6, 0], [-8, -2, -1, 6, -1, 1, -8, 1, -4, 3, 3, -3], [-5, 2, 1, 7, 1, -2, 3, -2, -1, -3, -3, 3], [8, -8, -4, 2, -4, -2, 6, 0, 4, -4, 0, 0], [5, 5, 3, -4, 5, 0, -3, 2, 1, -2, -4, -3], [-2, -7, -2, -1, 1, 0, -3, -3, 3, 0, -1, -2], [4, -14, -4, 3, -6, -9, 4, -2, -2, 2, 2, -6], [-16, -2, -1, -5, -4, 2, 2, 0, 0, -2, 0, -6], [19, -12, 3, 5, 1, -1, 6, -5, 3, 2, 2, 1], [-2, 15, 5, -2, 5, 9, 1, 3, -1, 0, 5, 4], [2, -4, 3, -4, 3, 0, 7, 1, 4, -5, -2, 2], [27, 7, 4, 3, 5, 3, -5, -1, 0, 5, -2, 2], [-1, 4, -4, 1, -1, 0, -3, -3, 3, -2, 1, 0], [8, -8, -1, 3, -6, -2, 8, -6, 2, -2, 0, -2], [20, -3, -1, -6, 4, 10, 1, -2, -5, -3, 8, -3], [-14, 0, -1, 4, -2, 5, -1, 4, -6, 0, 5, -3], [2, -6, -5, -4, -1, -3, 2, -5, 2, -1, -3, -5], [-7, -8, -6, 1, -6, -8, -1, 1, -1, 6, -1, -6], [14, -10, -2, 4, 0, 2, 6, 0, 2, 0, 4, -2], [0, -3, -5, -1, 0, -2, -7, -2, -3, 1, -5, 3], [-14, 0, -1, 6, -5, -3, 4, 3, 0, -3, -1, -3], [-5, -7, 2, 0, 4, 2, 3, 1, 3, -1, 0, 6], [1, -5, 3, 4, 0, 2, 0, -4, 0, 0, 4, -1], [3, 0, -2, -4, 5, 2, -2, 2, 2, -3, 0, 0], [8, -3, 3, 6, 0, -2, -1, -6, -1, 3, -2, -1], [-8, -8, -1, -4, -2, 1, 3, 0, -6, 0, 9, -1], [4, 10, -3, 0, 5, 2, 0, -3, 0, -3, -2, 1], [-5, 13, -2, -2, 1, 0, -2, 3, -4, 1, 2, -4], [34, -1, 1, -8, 4, 6, 4, -1, 3, -2, 4, -3], [-8, 13, 1, 6, 0, -2, -3, 6, -1, 5, 2, 3], [-13, -4, -4, 1, -3, -7, 7, -3, 1, 0, -4, 0], [20, 1, -1, 4, 5, -1, -13, 1, -3, 2, -3, 0], [6, 0, -2, 4, -2, -4, 4, -4, 0, -4, -4, -2], [5, -6, 1, 4, 0, 0, -2, -1, 0, 4, 0, -3], [-4, -7, -6, -6, -4, -2, 0, 4, -7, 1, 8, -6], [-3, 6, 6, 0, 4, 4, 2, 1, -2, 0, 4, 4], [-9, 1, 2, 1, 2, 4, 4, 4, 0, 0, -1, 4], [4, -11, 1, -4, -8, -4, 3, -2, -1, 5, 4, -3], [-14, -2, -5, -7, -5, 2, -2, 3, 0, 1, 0, 1], [4, 5, 1, 2, -1, -2, 1, -7, 1, 4, 2, 3], [-4, -9, -4, 0, 1, 0, 3, -7, 7, -2, 0, -2], [18, 0, 4, -7, 3, 4, 4, -7, -4, -3, 0, 1], [6, -4, -3, -3, -3, 0, 2, 4, 0, 0, -1, 2], [3, 5, 3, -6, 3, -2, 10, 1, 4, -1, 2, 1], [-7, -8, 3, -3, 1, 0, 1, 0, 3, 3, 3, -3], [-8, -7, 3, 3, -3, -7, -1, 1, 5, 2, -3, 1], [11, 0, -2, -12, 4, 6, 4, -9, 4, 0, 2, 3], [1, -3, 6, 4, 3, 3, -3, 2, 1, 2, -1, 1], [8, 3, 0, -3, -2, -5, -5, 4, -1, 1, -1, 0], [16, -1, 5, 9, 6, 4, 3, -2, -1, -1, 3, 3], [-6, -1, 2, 0, 3, 3, 1, -7, 5, 4, 3, 0], [-24, -9, -4, -3, -8, -6, 1, 5, -3, 0, -3, -6], [8, 5, 2, -5, 2, 0, -2, -7, 2, 0, -5, 2], [18, -9, -5, -2, 0, 0, -3, -2, -1, -3, -2, -1], [-12, 5, 1, 10, 4, 2, -3, 6, -5, 3, 0, 0], [6, -4, -1, 0, -4, 2, 13, 0, -2, -4, 2, -5], [6, 3, 5, 2, 4, -2, 3, 8, -1, -3, 0, 1], [4, -6, -1, 1, -3, -1, 2, 4, 6, 0, 1, -2], [-16, -5, -4, 2, -3, 2, -3, -3, -1, -6, -2, 2], [16, 12, 6, 0, 0, 3, -2, 6, -4, 2, -3, 4], [1, -8, 2, -6, -3, 2, 4, -7, 0, -2, 2, -5], [12, 11, 4, -6, 4, 4, 4, 0, -1, -5, -2, -2], [-4, -4, 3, 1, 2, 1, 2, -2, 2, 6, 0, 4], [-17, 6, 2, 2, 3, 2, -2, 9, -2, 2, -2, 5], [22, -5, -1, -6, -2, 4, 1, -6, 9, 1, -4, -1], [13, -13, -4, -1, -2, -3, 0, -4, -2, -2, 4, -2], [-20, -6, -2, -12, -5, -5, 2, -3, 4, -3, -1, -2], [8, 3, -2, 6, -3, -6, 3, 1, -3, 0, -2, -2], [-10, 0, -3, 2, 0, -6, -2, -5, -2, 3, -2, 1], [6, 7, 3, -5, 2, 2, -3, -2, -1, 3, 4, 1], [23, 3, 0, -2, 3, -1, 4, -1, 4, 3, 0, 4], [6, 11, 2, 14, 0, 2, -7, 6, -7, 3, 2, 2], [-26, -5, -1, -5, -1, -2, 5, -2, 3, 5, 1, -1], [17, 11, 5, -5, 3, 6, 0, 9, -2, -1, -1, 1], [-5, 10, 2, 5, 2, 0, -3, 5, -5, -2, -3, -4], [-6, -2, -1, -6, 2, 7, 4, 0, 0, -4, 3, 4], [-14, 10, 2, -4, 0, 2, 6, 0, 0, -8, -2, 2], [10, -7, -6, -16, -2, 2, -1, -4, -1, -1, 6, 1], [-2, 16, 5, 7, 6, 7, 0, 1, -4, -1, 1, 5], [4, -7, 0, 2, 1, 8, 5, 1, -3, 2, 6, -4], [-9, -5, -1, 4, -7, 0, 3, -3, 5, -5, 0, -2], [10, 0, -1, 3, -3, 3, 2, -5, -6, -1, 6, -3], [6, 4, 7, 2, 6, 7, 11, -6, 0, -4, 1, 7], [-12, -6, 0, 2, -7, 1, -2, 1, 0, -1, -3, -1], [20, -6, -1, -6, 0, 0, 0, -6, 2, 0, -4, -1], [19, -4, 4, 1, 4, 0, 1, -1, 7, -2, 3, 4], [-8, 7, 0, -4, 4, 2, -7, 0, -5, -1, 4, 0], [-6, -4, -1, -4, -1, -2, 0, -5, -2, -1, 2, 1], [-26, 2, 0, 5, 1, -6, -4, 8, 0, 2, -3, 4], [-18, 2, 0, -8, 1, -2, 6, -2, -2, -6, 0, 4], [-7, -2, -4, -1, -4, -2, -6, 4, -7, 3, 1, -6], [4, 5, 0, 0, -1, -6, 5, 3, 5, 4, 4, 2], [2, -1, 3, -4, -2, 2, -7, 2, 3, -1, -4, -1], [-8, 3, -3, 6, 2, 5, -9, 2, -3, -1, -5, 1], [6, 13, 3, -6, 11, 0, -3, 1, -3, 2, -4, 1], [8, -4, -4, 0, -1, -4, -9, 9, -4, 1, -6, -2], [-11, -3, -5, 3, -1, 0, 2, 1, -4, -3, -1, -5], [24, -9, -1, -8, 0, -1, -5, 2, 3, -1, 1, 3], [-7, 1, 1, -8, -3, 3, 2, -1, 4, -3, -5, 3], [8, -9, 1, -2, -1, 3, 9, -5, 9, 0, 1, -2], [-6, -11, -2, -3, -7, -6, 5, -9, 5, -2, -4, -2], [-10, 5, 4, -6, 1, 7, -5, 7, -1, -4, 1, 2], [-9, 0, 0, -7, -3, -1, 5, -3, -1, -2, 0, -6], [21, 1, -3, -2, -1, -6, -3, 10, -7, 6, 8, -7], [4, -8, -6, 8, -6, -7, 4, 2, -2, 2, -1, -2], [-20, -2, -5, -4, 1, -3, -1, -7, 4, 1, 5, 1], [22, 6, 6, 6, 4, 6, 2, 8, 0, 0, 0, 2], [-18, 4, -1, 6, -7, 0, -2, 3, 2, -7, 0, -5], [2, 1, 0, 8, 2, 2, -7, 6, -7, 5, -2, -4], [-21, 1, -2, -4, -4, -5, -1, 1, 1, -1, 1, -6], [-38, -5, -3, 6, -4, -5, 1, -2, 5, -3, -9, 1], [-8, 7, -1, 1, 4, -3, -1, -8, 1, 3, -3, 1], [-10, 7, -3, 7, 4, 4, 1, 4, -1, -3, -2, -1], [20, 6, 3, 3, 4, 2, -7, 5, -1, 1, 3, -1], [-9, 15, 0, 4, 2, 1, 0, 8, -6, 2, 2, -6], [19, 3, -2, -2, 2, -3, 7, -11, 5, -3, 1, 0], [11, -6, -5, 4, -5, 0, 10, -1, 0, -4, 2, -1], [-22, 5, -1, -1, 3, -2, -7, 1, -3, 4, 3, -5], [13, 5, -1, -7, 2, 4, 7, 0, -2, -6, -1, 1], [16, 12, 1, -3, 9, 9, 0, 0, 0, -2, -1, 2], [20, -1, 1, 7, -2, 4, -5, -4, -3, 1, 0, 1], [6, 4, 8, 12, 2, -1, 0, 2, 0, -2, -5, -2], [8, 6, 4, -10, -1, 4, 6, -1, 4, -1, -6, 4], [-2, 11, 1, -2, -1, -1, 2, -3, 5, -4, -5, 3], [11, -14, 1, 0, -6, 0, -4, -1, 2, 2, -4, 1], [-16, -2, -7, -9, -3, -3, -2, 9, 2, 3, -2, 1], [-6, -10, -5, 0, 0, -2, 0, 5, 0, 3, 4, -3], [-10, -2, -4, 0, 4, -1, 0, -2, -4, 2, 1, -2], [22, 2, 5, 7, -1, 4, 2, -5, -4, 1, 2, 1], [0, -10, 1, 1, -10, -6, 0, -3, 6, 3, -1, 3], [-2, 9, 7, 3, 6, 2, -1, 0, 1, 3, 0, 1], [-9, 13, 4, 3, 2, 1, 1, -2, 2, 0, -2, 4], [26, -5, -4, -2, 1, 3, 3, -7, 1, -2, 5, -4], [19, -7, -6, -4, 0, 4, 0, -2, 0, -6, 1, 2], [-14, -5, -3, -2, 4, 1, -7, -4, -5, -3, -3, -1], [2, 8, 3, -1, 6, 6, -2, -6, 4, -4, -5, 8], [2, -18, -4, -4, -10, -9, 6, 0, 4, 4, 3, -4], [-10, -2, 0, -4, -4, -5, -2, 6, -2, 8, 1, -6], [-14, 8, -1, -7, 5, -2, -4, -1, -2, -5, -1, 1], [12, -1, 3, 6, 4, 0, 1, 4, -3, 1, -4, -1], [52, -3, -1, -4, 0, 2, 3, 6, -3, 3, 4, -3], [-4, 3, 1, 5, 0, 1, 0, 3, -6, 0, 6, -1], [21, -1, 4, -7, 2, 2, 2, -2, 6, -2, -5, -2], [12, -13, 2, -6, -3, -7, 1, 1, 3, 0, 3, 0], [-19, 4, -5, -7, -1, -4, -7, 10, -3, 1, -5, -1], [-11, -2, 0, -8, -7, -7, 4, -4, 4, -3, -1, -2], [19, 4, 0, 4, 5, 0, 1, -9, 1, 0, 1, 2], [2, 7, -1, 4, 5, -2, 0, 3, -1, 0, -2, 1], [-11, 4, 4, -3, 1, 8, 1, -1, -3, -2, -1, 4], [-2, -8, -3, -6, -3, -4, 0, 1, 2, 3, 6, -3], [8, -7, 0, -8, -2, -8, 3, 3, 1, 2, 0, 0], [28, 3, 2, 6, 6, 5, -7, -6, 3, 1, -9, -1], [2, -1, -5, -2, -6, -8, -7, -2, 1, -3, -4, -5], [-19, -11, -2, -6, -2, -7, 8, -6, 10, 0, -4, 0], [-2, 2, 3, 4, -5, -2, 2, -3, 0, 7, -4, 3], [10, 7, -2, 10, -2, -4, -3, 4, -3, -3, 0, -1], [8, 1, 0, -8, 0, -3, 4, -7, 1, 4, -1, -5], [16, 10, 5, 3, -1, 7, 3, -10, -7, -4, 2, 1], [3, -6, 1, -12, -7, -6, -1, 1, -1, 0, 0, -5], [2, 1, -3, 12, -5, -6, -6, 3, -3, 2, 2, -1], [8, -9, 0, 10, -6, -3, 18, -4, -1, -1, -1, 2], [-2, 6, 2, 2, 4, 8, -2, 2, 0, -2, 0, 6], [33, 8, 3, 2, 10, 3, -5, 0, -1, 1, -5, 1], [-4, -3, 1, 3, 2, -5, 1, 2, -5, 1, -5, -5], [22, 9, 4, 2, 5, 8, -3, 5, 1, 2, 0, 8], [18, 7, 6, 0, 5, 1, 6, 5, -3, 4, 7, 2], [-6, -7, -2, -14, -5, 0, 9, -4, 3, -3, -2, -3], [29, -2, -3, -1, -2, -1, -1, -4, 3, 3, 4, 1], [0, 18, 2, 2, 8, 7, -3, 8, -8, 2, -1, 2], [-10, -15, -3, 0, -2, 0, -1, -4, -5, -1, 4, -4], [-10, 10, 3, 8, -3, -4, -7, 3, -2, 7, -2, 4], [25, 9, 2, 2, 4, -2, 0, -2, 2, -2, -7, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6672_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6672_2_a_bs();".
// 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_6672_2_a_bs(:prec:=12)
    chi := MakeCharacter_6672_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_6672_2_a_bs();".
// 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_6672_2_a_bs( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6672_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-32, -1296, 9624, -13024, -9046, 8874, 2043, -2202, -50, 223, -20, -7, 1]>,<7,R![34816, 18496, -54400, -23848, 28220, 9416, -7007, -1533, 884, 100, -51, -2, 1]>,<11,R![64, -137816, 214060, 158217, -337637, 92929, 33156, -15103, -320, 695, -44, -10, 1]>,<17,R![40278784, 47960192, -18014240, -12171520, 3089104, 1080140, -232762, -44003, 8510, 837, -149, -6, 1]>],Snew);
    return Vf;
end function;