
// Make newform 2667.2.a.k in Magma, downloaded from the LMFDB on 02 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2667_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_2667_2_a_k();".
// 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_2667_2_a_k();".
// 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 := [57, -121, -170, 313, 215, -247, -112, 88, 25, -15, -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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [2, -4, -1, 1, 0, 0, 0, 0, 0, 0, 0], [-51, 190, 72, -399, -59, 255, 8, -66, 5, 6, -1], [-45, -114, 255, 308, -246, -209, 110, 55, -24, -5, 2], [146, -402, -174, 840, -19, -520, 81, 128, -29, -11, 3], [193, -401, -431, 882, 272, -559, -37, 139, -10, -12, 2], [-204, 515, 330, -1162, -89, 761, -66, -194, 34, 17, -4], [191, -729, -9, 1743, -333, -1172, 279, 304, -82, -27, 8], [321, -829, -377, 1939, -180, -1325, 298, 352, -99, -32, 10]];
    Rf_basisdens := [1, 1, 1, 1, 7, 7, 7, 7, 7, 7, 7];
    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_2667_a();"
function MakeCharacter_2667_a()
    N := 2667;
    order := 1;
    char_gens := [890, 1144, 2416];
    v := [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_2667_a_Hecke( : Kf := Kf)
    return MakeCharacter_2667_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, -1, 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, -1, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0], [-2, 1, -1, 0, 0, 0, -1, 1, 0, 0, 0], [-2, 1, 0, 1, 1, 0, 1, 1, 0, 0, -1], [-1, -2, 1, -1, -1, 0, 0, -2, -2, -1, 1], [-1, 2, -1, 1, 0, 0, 0, 1, 1, -1, 1], [0, 1, 1, 0, -1, -1, -1, 0, 0, -1, 0], [-1, 0, -1, 1, -1, -2, 1, 1, 2, 0, 0], [-2, 0, -1, 0, 1, 0, 1, -2, 0, -1, 0], [1, -1, 1, -1, -2, 0, 1, 1, 0, 1, 0], [-3, 2, 2, 0, 3, 1, -2, 2, 2, 2, -2], [-1, 0, 1, -1, 1, 1, 0, 0, 1, 0, 1], [-2, 2, 0, -1, 1, -1, -2, 2, 2, 1, -2], [3, 0, -1, 0, -1, 1, 1, -1, -3, -1, 0], [-1, 0, -1, 1, 1, 0, 1, 1, 0, 0, 0], [-2, -2, 1, -1, -1, -1, 3, 1, 1, 2, -2], [-1, -2, 0, 1, -1, 0, 0, -4, -3, -3, 0], [-3, -2, 0, -2, 2, 0, 0, 3, 2, 3, 0], [-1, 0, -3, -1, -1, -1, -1, -1, -1, 1, -2], [2, 1, 0, -3, 0, 2, -2, -2, -1, -1, 0], [0, 0, 0, 0, -2, 0, -1, 1, 2, 1, 3], [-4, -2, 0, -1, -1, 1, 1, -1, 0, 0, 1], [-2, -2, 5, 2, 2, 2, -1, -3, -1, -3, 0], [0, 3, -2, 2, 0, 0, -1, 2, 1, -2, -1], [-4, -4, -1, 2, -1, 2, 3, -2, -4, -1, 2], [-7, -3, 0, 2, -2, 2, 1, -2, -3, -1, 3], [0, 0, -1, -4, -1, -3, -2, 1, 1, 1, -3], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 3, -3, 1, 3, 3, 5, -3], [3, -1, -1, 2, 0, -2, 2, 1, 0, 2, 1], [-4, 0, 1, 2, 0, 2, -3, -1, 1, 1, -2], [-2, -5, 1, -5, -3, 2, -1, -3, -1, 0, 2], [-7, 2, 0, 1, 2, -2, 1, 2, 0, 2, -3], [-2, 3, 1, -2, 4, 0, -3, 1, 0, 0, -4], [-1, 0, 2, 0, 3, 1, 2, -1, 0, 0, 0], [-4, 3, 2, -1, 6, 4, -4, 1, 1, 1, -4], [-6, 0, 0, 3, 3, -1, 4, 2, 0, 1, 0], [-2, 1, -4, 1, -1, 0, -2, -2, -2, -1, 2], [6, 3, -2, 1, -2, 0, -2, 3, 1, 3, 0], [4, -4, -3, 0, -4, -2, 0, 0, 1, 0, 3], [-2, 1, -2, -1, 0, -2, -1, -2, -1, 0, -1], [-5, 1, 4, -1, 0, 2, -6, 4, -1, 2, -2], [-4, -1, 2, 0, -2, 2, -2, -2, -1, -3, 2], [-4, -1, 3, 4, 2, 6, 0, -3, -4, -5, -1], [-2, -3, -1, 3, -2, 2, 5, -3, -2, -4, 6], [7, 1, 2, 4, 2, 2, 0, 0, -3, -4, 2], [4, 1, -1, -2, -1, 3, 0, -2, -6, -2, 3], [-2, 3, -1, 3, 4, 0, 2, 3, 0, 1, -3], [-5, 5, -4, 3, 5, -1, 3, 1, 5, 2, 1], [-2, -2, 4, 0, 0, 2, 0, 5, 4, 2, 2], [7, 1, 2, -8, 3, 1, -5, -2, -1, 0, -3], [-2, 1, 2, 1, 0, 4, 0, 0, -1, 1, 4], [-2, -3, -2, -6, -2, 1, -4, -2, 2, 0, 0], [5, -7, 1, 1, -2, -3, 5, -5, -1, -2, 2], [-11, -2, 5, 3, 2, 0, -1, 4, 1, 4, 0], [-4, 7, -4, -1, 0, 0, -4, -2, -1, -3, 0], [-4, -7, 0, 0, 1, 1, 5, -3, -1, -1, 1], [5, -4, -1, 1, -4, 0, 2, -1, -1, -3, 5], [-2, 1, -4, 2, -3, -3, 0, -1, 1, -2, 0], [2, -7, -2, 4, -1, -3, 6, -2, -5, 0, 2], [10, 6, -2, 2, 2, -2, -1, 5, 6, 1, -5], [-7, 2, 0, 1, 4, 4, 0, 1, 2, 1, -6], [3, -5, -1, 3, 1, -3, 4, 0, 2, -1, 5], [0, -4, -1, -8, -4, -4, -2, -1, 3, 6, -3], [-4, -8, 1, -1, -1, -3, 1, -1, -3, 0, 0], [-3, 5, 1, 3, -1, -1, 1, 2, 2, -2, 6], [-4, 3, -2, -4, -3, -5, -6, 3, 3, 2, -4], [-3, -4, 1, 2, 2, -3, 0, 2, 6, 4, -5], [4, 1, -2, 4, 3, -3, 1, -2, 5, -1, 3], [-2, 0, 1, -2, -1, -2, -2, -1, 6, 0, 1], [-4, 0, -2, 1, -3, -1, 2, 3, -4, 3, 0], [2, -1, 8, -7, 2, 2, -8, -6, -1, -1, -4], [14, 4, -5, 2, -4, -4, 2, 3, 1, 0, -1], [-4, 7, -2, 3, 3, 3, -1, 5, 3, 3, -3], [-8, -6, 4, -6, -5, 0, -3, -4, -1, 1, 1], [9, 3, -2, 6, -4, 0, 1, 1, -3, -7, 7], [-1, -2, 3, -1, 0, 2, -1, -2, -3, 2, -2], [3, 4, -4, 2, 4, 1, 1, 1, 1, -5, 1], [-3, -5, 1, 7, 1, -5, 6, 8, 8, 5, 3], [3, 0, 3, -3, -3, 3, 1, -1, -3, 3, 0], [-4, 9, 1, 3, 5, -4, 2, 6, 7, 1, -1], [9, -2, -2, -3, -2, -2, 5, 3, 0, 0, 5], [1, 6, -4, -1, -1, 6, -4, -5, -1, -5, 0], [-4, -4, -2, 1, -2, 4, 3, -8, -8, -7, 5], [-7, 0, 2, 7, 3, 3, 2, 2, 2, -2, 2], [-1, -10, 3, -3, -4, 4, -3, 0, -3, 2, 0], [4, 12, -2, -4, 1, 2, 0, 0, 7, 0, 4], [11, 1, 2, -4, 0, -2, -4, -4, 3, 0, -4], [9, -5, 0, 3, -8, 4, 4, -5, -5, -6, 6], [-16, 5, 1, 0, -1, 3, -5, -2, -4, -3, -4], [-1, -8, 0, 6, -7, -1, 7, -4, -4, -5, 7], [6, -3, 0, -5, 0, -4, 2, -9, -1, 3, -4], [5, 1, 6, 5, 2, 0, 7, 0, -1, -1, -1], [6, 0, -1, -3, 1, 1, -4, -1, 1, 3, -1], [8, 2, -2, 0, 1, -4, 2, -1, -1, 2, -4], [-2, -2, 6, 3, 2, 2, -3, -2, -2, 3, -1], [-4, -1, 5, -8, -5, 3, 1, -5, -6, -3, -2], [-4, -1, 0, 0, -5, -1, -3, -1, -1, -3, 1], [-1, 1, 8, 3, 3, -1, 0, -4, -5, -3, -2], [-2, 7, 3, -12, 3, 3, -13, 1, -2, 3, -10], [8, -5, -1, -2, 4, 2, -1, 5, 4, 2, -2], [1, 10, -4, -6, 1, -1, -7, 7, 8, 3, -3], [-7, -7, -3, 3, 2, -7, 9, -4, 1, 0, -2], [0, -14, 3, -5, -2, -1, 3, -1, -4, 2, 2], [1, 5, -9, -2, -4, -2, 1, 1, 2, -1, 0], [-6, 10, 2, 0, 0, -4, -5, -4, -2, -3, -3], [-5, 2, -4, 11, 4, 2, 3, 0, 0, -4, 3], [0, 2, 0, 7, 2, -7, 4, 1, 1, 1, 2], [-5, 2, -3, -6, -3, 3, 2, -4, -1, -6, 5], [0, 4, 0, 5, -5, 3, -3, 2, 0, -2, 7], [-3, 6, -6, 10, -3, -3, 5, 1, 2, 1, 5], [-5, -7, -1, 1, 1, -1, 5, -1, 2, 2, 2], [-2, -1, 2, 7, 4, -2, 8, 1, 5, -1, -2], [1, 8, -2, 1, 3, 1, -1, -1, 2, 3, -7], [7, -3, 2, -8, 0, -2, -8, 1, 5, 6, 0], [-1, 1, -2, -6, -3, 1, 1, -1, 1, 6, 1], [3, -3, 3, -3, -9, -7, 3, 4, 0, 0, 6], [-5, 17, -5, -2, -1, -5, -4, 3, 2, 3, 1], [-4, 3, -1, -4, -2, -3, -10, 1, -5, 0, -3], [-1, 11, -7, -3, 3, -7, -3, 10, 4, 2, -6], [-11, 0, 1, 1, 0, 4, 3, -2, -1, 6, 4], [10, -3, -5, -3, -6, -2, 2, 3, 0, 7, -3], [8, -8, -5, 0, -6, -6, 8, -5, -5, -4, 3], [8, -5, 0, -1, -1, -7, 3, -5, 1, 1, 1], [13, 1, -2, -6, -1, -2, -5, 1, 0, 3, -3], [2, -5, 3, 3, 0, -4, 5, -5, -4, -4, 0], [-1, 0, -10, -1, -4, -8, -6, 6, 2, 1, -2], [-15, 13, 4, 5, 2, 0, 4, 3, 3, -4, 4], [0, 8, -9, 4, 0, 0, 9, -3, 5, -1, 6], [-3, 6, -3, -5, -2, 4, -2, 1, -3, -1, 7], [8, 0, 5, -1, 0, 5, -2, -6, -6, -5, 1], [1, 1, 5, -4, 6, 0, 0, -3, 4, 2, -3], [-4, -3, 6, 5, 4, 0, -4, 5, 3, -3, 0], [-11, -1, -4, -2, -7, -2, 1, 3, -8, 1, 1], [-2, 4, -4, -2, -5, -3, 1, 7, 12, 8, 3], [12, -7, -4, -6, -9, 3, -7, -4, -7, -1, 1], [-3, -2, 4, -9, -3, -2, -1, 4, 5, 8, -5], [6, 0, -1, -4, -3, -3, -6, 5, 1, 1, 1], [10, 3, -4, -4, -1, 1, -9, 5, 7, -3, 1], [-12, -4, -6, 8, 7, -8, 6, 1, -1, 2, -10], [5, -1, 11, -1, 1, 1, 3, 2, 2, 0, 2], [-5, 1, 9, 5, -1, 9, -3, 2, -10, -4, 0], [11, 4, 8, -2, 9, 7, -3, 1, 4, -1, -5], [-4, 13, -2, 2, -1, -1, -1, 0, 3, -3, 1], [-8, -9, 3, 1, -4, 2, 2, 1, 2, 1, 3], [0, 7, 0, -4, 3, 7, 3, 1, 5, 3, 3], [8, -1, -2, 1, -2, -4, -1, -2, -5, 0, -3], [-2, -8, 5, 2, 3, 0, -3, -3, -4, -3, -2], [8, -1, 12, 3, 6, 8, -6, 6, 3, -1, 2], [4, 0, 2, -2, -3, -6, 0, 15, 7, 10, 4], [-1, -11, 4, 4, 2, 2, 3, -5, -7, -1, 5], [-6, 11, -13, 1, 1, 5, -3, -5, -2, -7, 2], [8, -11, 8, -5, 2, 2, -4, 0, -5, -1, -6], [-8, -2, 5, 10, 3, 8, 0, -2, -2, -6, 1], [10, -3, 1, -7, 4, 2, 0, -4, -4, 1, 1], [3, 2, -4, 5, 0, -4, 8, -5, 0, -1, 4], [-19, 11, -7, 11, 0, -2, -3, 1, 8, -1, 0], [-3, -6, -4, 5, 2, -2, -5, 8, 4, 0, 1], [-2, -10, 7, 5, -4, 4, 3, -4, -9, -5, 0], [-1, 8, -4, 0, 5, 5, -4, 12, 4, 8, -4], [2, -2, -7, -1, -6, -12, 5, 7, 7, 9, 2], [-10, -9, 3, 6, -2, -2, 6, 4, 4, 3, 5], [-1, -3, -3, 0, 3, -5, -2, 6, 4, 13, -7], [-17, 11, 7, -4, 6, -2, -1, 9, 4, 11, -8], [-10, 3, 8, 3, -2, 8, 4, -6, -7, -1, 4], [3, 3, -4, -6, -3, -2, 2, 4, 4, 0, -4], [-12, 4, -9, 1, 4, 2, 2, 5, 3, 8, 3], [-5, -10, 4, 0, 3, 3, 9, -13, -8, -3, -1], [6, -10, -5, 0, -4, -4, 9, -7, -7, -3, 4], [2, 1, 0, 6, 2, -1, 1, 5, 0, -1, 3], [-4, -1, 6, -10, -6, 2, 2, -6, 7, 7, 4], [-2, 1, -8, 1, 0, -2, 5, -9, -7, -6, 1], [0, 2, 0, -11, -1, -3, -3, 6, 0, 2, -7], [2, -4, -2, -3, -4, -3, 1, -6, -3, 6, -1], [8, 4, 2, -7, -1, 7, -8, -1, 4, 1, -8], [0, -5, 0, 5, -9, 0, 14, -6, -10, -5, 8], [6, -2, 0, -4, -2, -2, -7, -11, -8, -3, -5], [0, 3, -3, -3, -5, 5, -12, 4, 0, -2, -1], [14, 4, 3, 4, 0, 2, -5, 3, -11, -11, 2], [3, -20, 0, -2, -13, -5, 9, 0, -4, 7, 3], [-11, 2, -8, 1, -3, -10, 5, 3, 13, 4, -1], [-10, 13, 1, -4, 2, -5, -8, 15, 5, 10, -7], [-15, 4, -5, 2, 5, -1, -8, -3, -5, -6, 1], [-8, 1, -1, 9, 4, -6, 8, 6, 10, 7, -1], [0, 15, 1, 6, 8, 2, -4, 2, 2, -7, -3], [5, 9, -4, 3, 4, -1, -12, 4, 0, -3, 0], [19, 0, -1, -12, -5, 1, -3, -6, -3, 1, 2], [6, -9, 4, 5, 2, 2, 2, -5, -3, -11, 12], [0, 3, 8, -10, 11, 5, -4, -3, 1, 0, -6], [-12, -6, 13, -7, 1, 1, 6, -12, -9, -7, -1], [-12, -5, 0, -2, -6, 2, -4, -5, -3, -1, -2], [-9, -7, 11, 0, 7, 11, 7, -6, -10, -4, 4], [-1, 7, 8, 0, 9, -1, 1, 8, 5, 12, -9], [-13, 14, 3, -3, 4, 10, -16, 6, 5, 1, -3], [-2, 0, 9, 9, 6, 3, 10, 5, -6, -3, 1], [-3, -19, 7, 5, -12, 1, 2, 1, 1, 1, 5], [12, 5, -4, -1, -4, -6, -10, 9, -1, -1, -6], [-17, -2, 10, 4, -1, 9, -4, -3, -2, -2, 8], [2, -3, 0, -8, 4, 6, 2, -1, -3, 7, -8], [-20, -3, 0, 3, 0, 10, -2, -19, -15, -15, 4], [-3, 8, -4, 2, -1, -3, 9, -12, -10, -9, -1], [-13, 6, 0, 5, 10, 4, 5, 0, 2, -2, -7], [-9, 2, -3, -1, -2, 0, -2, 3, 1, -1, 3], [-5, -4, 1, 3, 7, -5, 2, 3, -1, -2, -7], [2, -5, 2, 5, 4, 0, 7, -8, -3, -6, -1], [-6, -17, 6, -2, -6, -3, 7, -10, -12, -3, 7], [-14, -9, 5, -1, -7, 0, 4, -13, -1, -9, 13], [11, -17, 4, -7, -3, 5, 4, -8, -11, -3, -2], [0, 16, 1, -7, 6, 1, -4, -1, 2, 3, 1], [-10, -11, 6, -1, 2, 2, 3, -3, -11, 4, -3], [9, 4, -2, -7, -5, 0, -3, 2, 3, 4, -3], [19, -8, 7, -3, -5, -4, 1, 0, -6, -8, 4], [-12, -7, -1, -10, 9, -7, 0, 3, 14, 6, -3], [-2, -6, 1, -6, 6, -4, -6, 10, 5, 0, -5], [-7, 2, -2, 4, -9, -15, 4, 5, 12, 4, 4], [-10, -8, 7, -11, -1, -3, 4, -10, 3, -1, 3], [-12, 15, -2, -3, -3, -2, -3, 10, 2, 4, -5], [-9, 9, 8, 2, 4, 0, 4, 9, -1, 2, -6], [9, -3, 1, 9, -7, -7, 4, -3, -2, -5, 5], [-4, -16, 1, -4, 4, 8, 8, 3, 3, 10, -3], [-12, 1, 0, 12, -5, -9, 7, 10, 9, 3, 3], [0, 3, -12, -5, -9, -11, 2, 6, 9, 2, 0], [12, 2, -6, -9, 10, -6, -13, 12, 18, 5, -9], [-3, 9, -8, 8, 1, -3, 6, 7, 11, 3, 4], [-6, -6, 2, -7, 0, 7, -1, 0, -1, 4, 9], [19, -3, -8, 5, -14, -11, 2, 4, 2, -3, 4], [15, -3, 10, -11, -1, 1, -5, -2, -1, 6, -1], [-16, -4, 11, 4, 10, 4, -4, -12, -11, -8, -3], [-11, 2, 3, -7, 1, -9, -3, 6, 7, 9, -6], [-10, 1, 6, 1, 9, 3, -4, -7, -7, -12, -2], [-14, 8, 5, 10, 11, 4, -6, 9, 8, 8, -1], [16, 8, 6, 0, 6, -4, -8, 16, 8, 12, -4], [22, 8, 3, 10, -2, -2, -2, -4, -13, -12, 5], [-12, -9, 0, 8, 5, 3, -2, -7, -5, -6, -2], [-12, -8, 9, 0, -2, 4, 0, 5, 7, 10, 3], [-4, 9, -9, -12, -3, -9, -6, 0, 4, 6, -3], [-13, -2, 2, 18, 7, -5, 10, 5, 0, 4, -2], [-21, -4, 4, 6, 3, 5, -7, 7, -8, 1, -5], [5, -24, 7, -4, 3, 9, 5, -10, -1, 7, 4], [-4, 14, -2, 6, 4, -2, 0, 1, -6, -6, -10], [-27, 1, -2, 4, 5, 9, 10, -2, -11, -3, -4], [-22, 9, 5, -3, -2, 0, 0, -5, 4, 9, -3], [4, 2, 0, 0, 5, -5, 0, -12, -6, 1, -12], [1, 8, -6, -10, 5, -1, 2, 3, 10, 4, 0], [3, 6, -2, 5, 6, 8, 8, -9, -2, -3, 0], [-8, 9, -7, 8, -1, -11, 9, 3, 16, -1, 4], [-2, -11, -10, -2, 1, -1, 8, -5, -3, 6, 4], [-22, 2, 0, -3, 15, 7, 1, 4, 4, 6, -1], [-1, 5, 7, 2, -8, 0, -1, -4, -8, -9, 12], [-17, -14, 14, 9, 4, 2, 7, -7, 0, 0, 3], [20, -8, -2, -5, -11, -3, -1, -1, 2, 8, 3], [7, 5, 17, -2, 0, 2, -4, -11, -10, -10, -3], [5, 20, -3, 2, 9, -1, -4, 22, 9, 14, -3], [-6, 10, -1, -14, -4, -4, -5, -11, -3, -1, -2], [2, 13, -5, 2, -5, -3, -16, 4, -6, -4, 1], [-15, 8, -15, 0, 6, -3, -3, -3, 4, 5, -8], [-10, -11, 9, -1, 4, 4, 2, -6, -10, 1, 3], [3, 5, 2, 2, -2, 4, -11, 10, 1, -3, 5], [4, 3, -2, 6, 6, -10, 10, -1, 11, 7, -4], [-2, 6, -17, -5, -4, 1, -6, -4, 6, -1, 9], [4, -8, 5, -2, -7, 7, 1, -18, -11, -18, 8], [1, 13, -5, -14, 6, 4, -14, 4, 6, 6, -15], [28, 7, -4, -4, -5, -1, -3, -4, -3, -9, -5], [11, 11, -6, 11, 5, 1, -9, 5, 9, 0, -1], [14, 3, -1, -3, -9, 2, -3, -8, -5, -4, -4], [11, -4, -3, -7, -9, 2, 1, -19, -8, -16, 4], [-35, 17, 0, 10, 0, -2, 3, 5, 3, -1, 3], [-16, 0, -14, 3, -3, -1, -1, 1, 4, 6, 1], [6, 0, 2, -4, 4, 8, -10, 10, -4, -4, 2], [-15, -2, 5, 9, -4, 10, 7, -9, -11, -2, 2], [1, -7, 10, 5, 1, 3, 3, -7, -1, -8, 7], [-8, -12, 2, 2, -4, 2, 3, 2, 0, -7, 3], [7, 13, -3, 4, 6, -4, -5, 1, -6, -5, -10], [14, -7, -14, 0, -12, -2, 1, -10, 1, -6, 7], [4, -2, -12, -6, 5, -7, 5, -8, 8, -4, 1], [-1, -5, -10, 0, -10, -2, 0, 8, 1, 4, 2], [27, 0, 5, 1, 4, -10, -3, 7, 11, 2, 0], [19, -12, -6, 2, 1, -3, 3, 9, 4, 3, -7], [4, -4, -4, -11, 1, 7, 2, 8, 0, 7, 2], [2, 13, -3, -4, 10, 3, -10, 13, 17, 8, -3], [11, -5, -5, 5, -3, -5, 14, 1, 2, 5, 1], [-1, -19, -2, 2, -2, 4, 13, -15, -9, 3, 11], [-13, 0, 1, -2, -3, 3, -9, 11, 7, 7, 0], [4, 12, 3, 9, 5, -11, 0, 6, 3, -5, -1], [-23, 10, 12, 3, 16, 18, 6, -1, -4, 1, 4], [3, 0, -8, -1, -8, -2, 1, 2, 0, 2, 7], [-3, 25, -7, -11, 1, -3, -9, 0, 2, 4, 0], [-2, -13, 5, -2, 1, -1, -1, -14, -4, -5, 2], [10, 2, 6, 4, 1, -4, -10, 9, 1, -2, -8], [-1, -7, 11, 6, -1, -4, -6, 9, -3, 4, -7], [-2, 10, -19, -4, 1, -10, 0, 7, 4, 12, -9], [3, 37, -1, 1, 5, 3, -5, 3, 2, -2, -4], [14, -13, -1, -1, -9, -8, 20, -6, -9, -5, 7], [2, 5, 2, 6, -11, -1, -5, 8, 3, 1, -1], [-28, -12, 7, -2, 1, 8, 10, -14, -16, -4, 3], [10, 9, 5, 9, 7, 3, -3, 2, -2, -1, -2], [-15, 8, -14, 0, 10, -1, -3, -2, 1, 5, -1], [-6, 14, 0, 5, 4, 3, -6, 3, 7, -3, -4], [-4, 0, -8, -10, -2, -6, -12, 0, -2, 0, -2], [2, -5, 0, -12, -4, 1, 1, -12, 0, -3, 3], [23, 3, -3, -16, -1, -8, -4, 14, 11, 16, -3], [-1, 1, -4, 16, -8, -12, 6, -8, -1, -8, 4], [15, -7, -8, 2, 0, -2, 3, 0, -1, -9, -3], [2, 2, -2, -1, 2, 15, -9, -5, -3, -2, -7], [-2, 0, -2, -2, 10, 4, 5, -7, -6, 5, -5], [-5, -5, -4, 1, 11, 5, 1, -12, -9, 2, -11], [-18, -2, -3, 12, -11, -2, 3, 10, 6, -3, 12], [-14, 2, 5, -8, -4, 0, -8, -11, -3, -8, 3], [17, 31, -8, -10, 1, 3, -12, 2, 9, -1, -2], [11, -26, -6, -10, -7, -3, -2, -4, 4, 10, 4], [17, -1, 8, -8, 4, 12, -4, -2, -9, 2, 4], [-6, -6, 4, 7, 3, -5, 3, -3, 6, 0, -7], [-2, -7, -7, -13, -8, -6, -3, 9, 6, 14, 4], [-7, 4, 17, -1, 4, 6, -13, 14, 5, 0, -12], [0, 4, -5, 5, -2, -17, 2, 5, 4, 1, -5], [-1, -5, 3, -2, 4, 10, -3, -14, -14, -7, 2], [-4, 11, -13, 5, -8, -10, 6, -5, -12, -7, 7], [11, 16, 7, -10, 2, 9, -12, -2, 0, 0, 1], [20, 1, -6, -1, 4, 2, -3, 9, 9, -2, -3], [21, 2, -15, -2, -5, -11, 6, -3, 13, 4, 3], [-23, 5, 1, 3, 7, 5, 2, 1, 10, 9, -3], [0, -23, 5, 3, 0, 10, 0, -18, -12, -13, 9], [8, 14, 1, 11, 6, 3, 0, 2, 4, -15, -3], [3, -6, -1, 4, -1, -1, -7, 9, 5, -5, 4], [15, -25, 19, 10, 1, 9, 7, -3, -16, -6, -2], [6, 12, -4, 9, 11, 9, -2, 15, 4, 9, -12], [-11, 1, -3, 10, -2, -12, 7, 5, -2, 7, -10], [-5, 2, -16, 7, 3, 2, 18, -3, -5, -7, 6], [-8, 1, 8, -10, 5, 7, -10, 5, -7, 4, -12], [10, -6, 1, -4, -18, -8, 6, 15, -3, -2, 13], [18, 13, -2, 18, 3, 1, -1, 1, 1, -11, 1], [7, -18, 5, -8, -13, 9, -5, -12, -7, -11, 18], [15, 3, -13, 1, 3, -3, 3, 13, 18, 0, 4], [-2, -13, -2, 4, -3, -9, 1, 10, 21, 15, 3], [-3, -2, 0, -6, -9, 3, 8, -16, -18, -8, 6], [29, 6, -8, -7, -7, -11, -4, 4, 8, 4, -10], [6, -8, -5, 8, -1, 3, -3, 3, 7, 4, 8], [-7, -20, 0, -7, -8, 2, -6, -5, -6, 3, -6], [-15, 2, -1, -2, 0, -8, 4, 6, 1, 15, -5], [-1, -12, 7, 0, 3, 1, 15, -10, -1, 5, 0], [-11, 4, -25, 2, -6, -8, 6, 0, 1, 9, 5], [-16, 11, -7, -5, 7, -1, -14, -12, -6, -4, -7], [-2, -3, -4, 0, 1, 3, -3, 3, 5, 1, -5], [-37, 5, 0, -1, 13, 15, -4, 1, 1, -1, -10], [22, -21, 20, 4, 2, 6, -2, -12, -5, -9, 8], [9, 4, -4, 4, 9, 3, -12, 0, 0, -4, 0], [-5, 3, 5, -4, 6, 0, -3, -6, -8, -9, -4], [-22, -1, -9, -9, 4, 0, -4, -4, 0, 11, -3], [3, -8, -7, -2, -13, -1, 3, -10, -7, -13, 10], [12, 3, 4, -9, 10, 12, 2, -13, -7, -3, -14], [-7, 14, -10, 2, 5, 1, 1, 31, 14, 17, 7], [-13, -2, 10, -8, 10, 0, 1, 11, -4, 8, -9], [-11, 17, -12, 12, 6, 0, 9, -11, 3, -13, 5], [17, 8, 2, 4, 4, 0, 20, -7, 0, -5, 2], [17, 10, 3, 3, 4, 6, 3, 5, 7, 4, -8], [6, 13, -10, 0, 4, 5, -4, -14, 2, -2, -8], [7, -2, -1, 4, -1, 5, 2, 1, 5, -4, 3], [12, -2, 6, 10, -2, 6, 3, 5, -2, -11, 9], [0, 7, 14, 11, 10, -2, -1, -2, -1, -4, -15], [15, -24, -4, -2, -8, -10, 1, 8, -4, 0, 5], [35, -3, 0, -10, 4, -6, -6, -2, 3, 6, -10], [12, 7, -6, 20, 3, -5, 0, 1, -9, -10, -2], [30, 10, 7, 7, 9, 3, -1, 4, -1, -2, -2], [-1, -13, 13, 4, 7, -4, 12, 9, 13, 8, -1], [23, 16, -7, 4, -7, -17, -4, 17, 13, 10, -11], [-9, 7, -10, 13, 7, -9, -3, 0, 1, -2, -15], [-23, -9, 6, 3, -5, -1, 4, -6, 1, -5, 4], [15, -16, 4, -1, 0, -10, 9, 7, 2, 12, -7], [4, -13, -8, 0, -5, -17, 1, 2, -1, -1, -15], [-2, 17, -14, 13, -3, -9, 0, 16, 17, -4, 2], [9, 12, 5, -4, 0, -7, -13, 6, 2, 3, -6], [4, -3, 9, 12, 8, -6, -1, 0, 6, -2, 6], [14, -3, 3, -15, 0, 6, -11, 1, 2, 8, -10], [18, -3, -4, 2, -9, -9, 13, -18, -13, -13, 3], [-6, -11, 23, 3, 6, 14, -8, -2, -2, -3, 13], [-7, -17, 3, -6, -9, 3, 0, -12, -2, -3, 19], [13, -24, 8, 9, -4, 0, -3, -6, -2, -4, 5], [4, 13, -21, 2, 0, -4, -6, 0, 12, -11, -7], [-16, -16, 4, 7, -5, 15, 7, -2, -14, -12, 5], [7, -10, 7, -2, 4, 8, 3, -13, -5, -2, -6], [-10, -4, 9, 17, -3, 1, 5, -1, -11, -2, -6], [33, -16, -7, -11, 2, 8, -1, -14, -3, -6, 6], [17, -16, 11, 0, 7, 7, 12, -18, -9, 2, -1], [10, -6, 8, -13, -6, 0, -12, -1, 2, -4, -4], [18, -30, 21, -2, -4, 4, 13, -9, -17, -7, -4], [8, 21, -6, -4, 8, -11, -10, 14, 18, 4, -6], [24, 16, -14, 2, -4, 0, -13, 8, 4, -7, -1], [25, -23, 8, -1, -6, 7, -6, -12, 2, -1, 14], [-21, -8, 2, -11, 7, 6, -10, 11, 9, 15, -2], [8, -1, 8, -2, -10, -1, -2, 13, 8, 6, 0], [4, -9, -11, -8, -3, 3, 5, -2, 10, 5, 4], [4, 0, 5, 16, 7, -2, 8, 13, 4, -4, -3], [2, -8, 2, -2, 2, 14, 2, -3, -8, -2, -2], [0, 18, -8, -10, -3, 2, -12, -8, -5, -18, 0], [19, -18, 0, -3, -1, 17, 0, -4, -8, -2, 0], [28, -11, -4, -17, -3, -8, -7, -24, 0, -2, -7], [0, -1, 3, -3, 1, -22, 6, 11, 11, 11, -5], [-15, 23, -12, 3, 8, -5, -18, -7, 10, -9, 0], [6, 24, -4, 1, 13, 1, -10, 6, 12, 1, 4], [8, -21, -2, -9, -7, -3, -3, 6, -5, 15, -1], [6, -3, 21, -1, 2, 6, -1, 16, -4, 2, -4], [8, 12, -23, -6, -3, -7, -10, 10, -5, 7, -5], [-14, -1, -2, -13, -5, 8, -11, 8, -2, 6, -3], [7, -16, 0, 16, -1, -5, 2, 14, -2, -2, 0], [6, -20, -4, -7, 5, -7, 6, 1, 4, 7, -10], [25, -4, -3, -11, 1, -2, 4, 12, 0, 13, -11], [27, 4, -7, -8, -6, 5, -7, -1, 10, -7, 0], [-4, -2, -7, -8, 2, -4, 8, 6, 13, 6, -3], [-22, -3, -1, 7, 4, -6, 3, 5, 2, -4, 4], [8, 17, -3, 15, 7, -3, 0, 21, 12, -2, 13], [4, 0, 0, 1, -5, -5, 4, 0, 16, 1, 8], [8, -10, -4, -15, -3, 7, 9, -12, 0, 4, -1], [-7, -16, 17, 2, -7, 3, -6, -8, -19, -2, -1], [5, 20, -7, 10, 11, -15, -3, 29, 17, 13, -6], [8, -15, 11, -2, 7, 11, 5, -12, -6, -9, -10], [-6, 19, -2, 4, -6, 4, -10, -2, -7, -13, -6], [3, -25, -1, -4, -4, 6, -4, -4, -4, 8, -3], [-23, 13, -17, 6, -4, -12, -7, 1, -2, -11, -6], [-12, -15, 6, -12, -5, -3, -3, -4, 11, 13, -1], [30, -20, 14, -8, 2, -4, -8, -4, -6, 6, 0], [-10, 10, -15, 18, 9, -2, 11, 22, 18, 19, 6], [-12, -12, 3, -11, -6, 11, -1, 4, -2, 4, 6], [-16, -8, 7, 13, -8, -4, 13, 14, 7, 13, 2], [0, -14, 16, 0, 1, 1, 3, -7, 4, -10, 11], [-13, -9, 0, -1, -19, 5, 1, -1, -19, -6, 13], [-2, 10, 14, 20, 15, 17, 7, -11, -4, -20, -1], [25, -4, -6, 11, 4, -14, 8, 9, 4, 11, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2667_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2667_2_a_k();".
// 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_2667_2_a_k(:prec:=11)
    chi := MakeCharacter_2667_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_2667_2_a_k();".
// 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_2667_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2667_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-57, -121, 170, 313, -215, -247, 112, 88, -25, -15, 2, 1]>,<5,R![-288, -400, 1192, 2212, 22, -1331, -197, 319, 32, -32, -1, 1]>],Snew);
    return Vf;
end function;