
// Make newform 2667.2.a.j in Magma, downloaded from the LMFDB on 05 May 2024.
// 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_j();".
// 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_j();".
// 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 := [-4, -13, 7, 20, -3, -9, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [5, 7, -4, -7, 0, 1, 0], [4, 11, -3, -8, 0, 1, 0], [2, -4, -12, 3, 8, 0, -1], [4, 22, 6, -18, -8, 2, 1], [3, 16, 15, -14, -15, 1, 2]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 1];
    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)
    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+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, 0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0, 0], [-1, 0, 0, -1, 0, 1, 0], [1, 0, 0, 0, 0, 0, 0], [-1, -1, 0, 1, -1, 1, -1], [-3, 0, -1, 0, 1, -1, 1], [0, -1, 0, 1, 0, -1, 2], [-1, -2, -1, -1, 0, 2, -2], [1, 3, 1, 1, -2, -2, 1], [-1, 0, 1, -1, -2, -3, -1], [-5, 1, 1, 1, -1, -1, -1], [-5, -3, 0, -1, -1, 2, 0], [0, 0, -2, 1, 1, -1, -2], [-1, -1, 2, 0, 0, 1, -3], [3, 1, -5, -3, 2, 0, 0], [0, -1, 3, -1, 0, 3, 0], [-6, 1, 2, 4, 0, 1, 3], [-3, 1, -1, -1, -1, 3, -1], [-3, -1, 1, -3, 2, 1, 0], [2, 1, -5, -1, 1, -3, 1], [-3, 5, 0, 0, 0, -1, 2], [1, 0, 1, 0, -4, -5, -2], [-5, 3, -2, 2, 1, -3, 2], [0, -1, 2, -2, 1, 1, -7], [0, 4, 1, 1, 1, 2, 5], [-5, 2, 4, 2, 0, 0, 0], [3, -3, 1, -7, 2, 3, 0], [1, 1, 4, -2, 0, -4, -1], [-3, 10, 3, 6, 2, -3, 4], [3, 3, -2, 5, 4, -1, 3], [-1, 0, 0, 0, 0, 0, 0], [3, 7, -1, 4, 3, -7, 4], [4, -8, -1, 0, 4, 3, -2], [-9, -2, 4, 1, -5, -2, 3], [-5, 0, 2, 3, -7, 0, 3], [-8, -1, 3, 5, 2, -2, 1], [1, 2, -7, -7, 2, 1, 0], [-8, 1, 2, 5, -2, -4, 7], [-6, 3, 4, 1, -3, -2, 4], [-6, -5, 1, -8, -1, 9, -5], [8, 0, -4, 2, 8, 8, 2], [-4, -5, 8, 3, -3, 0, 2], [-1, 7, 0, 4, 5, 1, 9], [-2, -4, -8, -2, -3, -5, -2], [3, 0, 2, -4, 2, -3, -1], [-2, 2, -3, 3, -2, -3, 8], [-4, -3, -3, 1, -2, 5, -2], [4, 5, -6, 0, 9, -1, 2], [-8, -11, -3, -3, 3, 5, -5], [-4, -3, 5, 1, -2, 5, -5], [-3, 8, 2, 0, -3, -1, -2], [0, 1, 0, -5, -3, -3, -7], [-1, -8, -4, 0, 3, 5, -7], [0, -6, 5, -1, -4, 4, -7], [-9, -9, -2, -4, -3, 5, 2], [4, -9, 1, 2, 5, -3, -1], [-6, -11, -2, -7, -3, 3, -8], [-11, 4, -5, -6, 2, 1, -1], [5, -11, -4, -1, 6, 5, -5], [5, -5, 7, 6, -7, -3, 0], [-1, -7, 3, -2, 1, 10, -6], [-2, 2, -7, 1, 14, -1, 5], [-1, 2, 5, -1, 3, -4, 4], [2, 1, 8, -4, -3, 9, -1], [0, -2, -1, 8, 6, -5, 1], [-1, -3, 0, 5, 7, 5, 3], [6, -9, -4, -12, -6, 4, -8], [3, 3, 3, 3, 4, 1, 2], [14, 2, 2, -1, 3, 1, -7], [-2, -6, -5, -7, 12, 9, -1], [-1, 7, -8, 5, 3, -3, 1], [12, 3, 5, -2, -2, 4, 5], [-1, -2, -8, -4, 5, 3, 2], [-9, 13, -1, 10, 0, -8, 0], [-1, -9, -6, -1, 0, 7, -3], [-8, 9, 0, -2, 4, -4, 4], [2, 8, -4, 1, 2, -1, 0], [-4, 8, 6, 5, 2, -11, 5], [-6, 8, 5, 7, -4, 2, 0], [3, 4, 7, 1, 3, -5, -4], [6, -11, -6, -10, -2, 5, -12], [-1, 11, 6, -1, -9, -9, 1], [9, 2, -5, -4, -2, 3, 10], [5, -1, 4, 2, -7, -9, -3], [-15, -9, -5, -3, 7, 11, 0], [11, -1, -9, -3, 2, -2, 0], [12, -16, -5, -14, 2, 14, -1], [-6, 10, -1, 4, 2, -9, -2], [-7, -12, 7, -7, -7, -1, -8], [-5, 3, -2, -4, -3, 1, -2], [2, 9, -1, 6, -2, -11, -2], [8, -10, -4, -8, 5, 10, -4], [-9, 2, 11, 4, -1, 3, -2], [0, -6, -4, 1, 5, 9, -2], [2, 9, -8, 7, -1, -4, 0], [8, 7, 10, 1, -3, 3, -2], [6, -14, -1, -10, -9, 7, -9], [0, 7, 10, 11, -13, -10, -2], [-4, 8, -3, 1, -7, -2, 0], [-8, 0, 1, 0, 0, -2, -5], [13, -9, -1, -8, 3, 13, -2], [2, -14, -14, -11, 6, 10, -6], [6, -10, -3, 0, -16, 0, -5], [-3, -6, 3, 5, -4, 1, 2], [-1, 8, -4, 5, 10, -2, 1], [15, 11, 8, 0, -5, -6, 4], [0, 4, -17, -4, 4, 0, -3], [-1, -11, 9, 11, 7, 3, 5], [-1, 9, 12, 2, -4, -4, 8], [0, 1, -9, 3, -6, -2, 1], [6, 8, -7, -6, 5, -3, 10], [-10, -6, 2, 0, -15, -4, 2], [9, 0, 11, -4, -13, 7, -7], [-5, -6, 8, -1, -6, -6, 3], [-5, -14, 4, -12, -15, 4, -11], [16, -3, -4, -5, 7, 2, 4], [-16, 5, 5, 12, -3, -10, 11], [-7, -3, 3, 4, -1, -4, 5], [-9, -2, 5, -7, -11, 5, -4], [4, -2, 10, -1, -9, 3, 1], [-5, 8, -9, 15, 14, -18, 3], [6, -4, 9, -3, -6, -3, -4], [-4, 8, -4, 4, 2, -9, 4], [5, -6, -11, -1, 11, 13, 0], [-19, 0, -2, -7, -6, -9, 5], [-4, 7, -10, 5, -3, -2, 3], [-4, -16, -8, -11, 4, 15, -6], [-9, -7, 3, 9, 5, 3, -6], [2, -1, 0, -6, -7, 11, -10], [-15, 6, -1, 7, -20, -8, -5], [2, -10, 14, -3, -2, 0, -9], [17, 6, 2, 0, -2, 2, -4], [3, -1, -9, -6, 13, 0, 14], [-8, 1, -17, 2, 13, -3, 0], [-10, 6, 5, -4, -14, -1, 10], [2, 1, -4, -5, -13, -10, -6], [7, -9, 7, -3, 1, 5, 6], [19, 1, 2, 3, 11, 4, 6], [6, -9, -11, -16, 4, 8, -5], [5, 0, 5, 6, 6, 1, 2], [-29, 9, 2, 2, -1, 11, 6], [5, -9, 3, 1, 2, -3, 5], [8, -9, -2, -9, -7, 12, 4], [0, -2, -20, -7, 3, -5, -1], [-12, -10, 6, 4, -6, 5, 8], [9, 4, 0, -4, 13, 2, 6], [-6, 3, 1, 10, 6, -4, 9], [5, 10, -2, 4, 7, 3, 16], [1, 3, 9, 0, -15, -9, -8], [-2, -4, -4, 7, -14, -3, -5], [-12, 1, 10, 8, -7, 5, 2], [-11, 21, 12, 8, 1, -11, 18], [18, 3, -10, -2, -4, -2, 11], [-12, 10, -3, -1, 6, -6, 16], [23, -1, -2, -2, 14, 0, -14], [15, 7, -2, 3, 12, 1, -1], [14, -6, 10, -8, 0, 20, -1], [14, 18, -6, 1, 3, -4, 16], [2, -8, -11, 8, 4, 9, -9], [14, 8, 2, -3, 3, -7, -10], [12, -3, 11, 3, -2, 4, -1], [9, -15, -11, -18, 5, 7, -12], [-7, 5, 0, 4, -10, -9, -9], [-16, 11, 7, 16, -4, -8, 23], [-15, -7, 7, 12, 2, 4, 4], [2, -2, 1, -6, 0, -9, 5], [9, 1, 5, 18, 4, -11, 8], [9, 7, -10, 5, 10, 7, 7], [-3, 14, 7, 15, -10, -12, -3], [11, -2, -1, -9, -1, 20, -20], [-1, 10, -2, 6, -7, -24, 12], [-15, -14, 10, 5, -5, 10, -5], [18, -4, 5, 3, -1, 0, -5], [-6, -6, 3, 1, -6, 5, -6], [-7, -10, 14, 6, -8, 0, 3], [-3, 0, -7, -12, -3, 13, -7], [-10, 1, -1, -11, 9, 16, -10], [-17, -5, 1, 2, -24, -2, 2], [12, 2, -4, 2, -6, -2, 4], [8, 1, -14, -15, 0, 15, -14], [-1, 8, -8, 9, 18, -7, 12], [-23, 14, -4, 6, 18, -12, 6], [-2, -10, -6, 3, 5, -3, 8], [-10, 4, 9, 2, -7, -14, 1], [8, 6, -15, -12, -3, -9, 1], [-1, -4, 8, 5, 5, 1, -6], [-8, 0, -12, -7, 9, 16, 6], [-9, 11, 17, 23, -3, -8, 3], [-13, -19, 8, 0, -6, 23, -7], [-17, 1, -8, 5, -8, -16, 9], [-12, -3, 0, 1, -19, -12, 0], [4, -12, -17, -19, 26, 21, -5], [-9, -18, 5, -4, -2, 4, -24], [-9, 12, 1, -4, 0, 10, -2], [-3, 9, -1, -3, 9, 5, 4], [6, 8, 9, -1, 8, 0, -1], [14, -5, -1, 6, 0, 5, 5], [17, 7, -13, -2, 6, -4, 5], [-3, 8, 0, -6, 6, -3, -11], [-10, -5, 2, -8, 4, 14, -15], [0, -23, -2, -19, -14, 14, -27], [7, -16, -1, -3, -5, -6, -9], [16, -5, 4, -2, -4, -2, -11], [4, 14, 2, -3, -3, -19, 6], [18, -9, -3, -18, -9, 18, -12], [-15, 7, 1, -12, -3, -6, 10], [-4, 9, 3, 0, 8, -5, 4], [0, 16, 16, 26, -19, -33, 22], [-5, 14, -1, 15, 2, -23, 20], [-12, -13, -8, -4, 2, 6, -16], [-2, 12, 4, 19, 0, -22, 20], [-19, 5, 3, -3, -21, -2, 2], [7, 16, 8, 10, 4, -14, 17], [3, -4, 5, -4, 18, -5, 6], [2, -7, -5, -18, -7, 18, -13], [11, 3, 11, 13, -11, -18, 14], [7, 13, -8, 12, 5, -15, 11], [-6, -11, 26, 11, -12, 0, -5], [15, -14, 5, -4, 0, 16, 4], [-21, 19, 6, -1, -1, -8, 15], [-9, 13, 5, -5, -12, -10, 6], [-32, -3, -18, 6, 15, -2, 3], [2, -13, -12, -15, -13, 0, -11], [4, 5, 13, 2, -14, -4, 5], [4, 7, 16, 12, -6, -14, 26], [28, -10, -25, -15, 13, 16, -10], [-4, 9, -3, 14, -2, -5, 3], [20, 5, -15, -16, 4, 0, -5], [-23, 11, 12, 10, 4, -15, 9], [-7, -3, 14, 17, 8, 6, 8], [3, 9, 8, 8, -5, -23, 9], [-6, -8, 7, -3, 10, 4, -22], [-3, -21, 10, -2, -13, 5, -7], [-11, -2, -10, -3, 6, 13, -7], [18, 10, 0, 16, 0, -2, -4], [-9, 29, -2, 7, 4, -17, -6], [20, -6, 1, 19, 0, -17, -3], [-24, -3, -8, -6, 2, 10, -2], [12, -7, -7, -11, 0, -1, 13], [-6, -7, -14, 11, 10, -2, -3], [23, -10, -8, -7, -6, 4, 3], [-11, 7, 16, 15, -24, -16, 7], [-5, -6, 2, -9, -2, -3, -12], [25, 19, 1, 5, 12, -10, 11], [13, 8, -10, 12, -1, -9, 18], [19, -20, -16, -9, 8, 7, -13], [8, -25, -2, 2, -15, 2, 0], [-30, -4, 3, -4, -8, -7, 13], [34, 0, 13, -8, -14, 8, -11], [18, -24, 1, 1, -4, -11, -13], [1, -20, 5, -22, -13, 17, -5], [0, -5, -3, 4, -7, 14, 5], [-9, -9, 7, -13, -3, -3, 0], [28, 20, 3, -1, 1, -22, 5], [-8, -8, -15, 2, 2, 19, -12], [-8, 20, 18, 16, -15, -36, 8], [21, 0, 4, 3, 1, 6, 17], [-4, -7, 1, -4, 11, -3, -6], [3, 22, -12, 10, 11, -18, 13], [-2, 2, -2, 7, 21, 6, 8], [1, -2, -17, -4, 22, 7, 4], [-1, 16, -17, -4, 12, -16, 10], [2, 1, -1, 1, -6, -7, -5], [-25, 10, -6, 11, 7, -14, 0], [11, -30, -11, 2, 21, 8, -21], [-18, 3, -9, 20, 0, 0, 21], [-22, 25, -4, 1, 3, 7, 9], [-22, -2, 16, -3, -4, -13, 10], [3, 13, 5, 14, -4, -21, 12], [-14, 8, -11, 8, 13, 14, 23], [12, -8, 3, 9, -3, 6, -1], [6, 18, -2, -12, 0, -12, -8], [14, 22, 5, -3, 2, -2, 1], [2, 3, -6, 2, -16, -15, -4], [1, 1, -6, 9, 15, 12, 15], [-19, 1, 13, 4, -8, 3, 14], [9, 9, -9, -9, 0, -3, 4], [10, 1, 8, -6, 19, 7, 6], [-28, -1, 7, 5, 9, 15, 7], [8, 10, -11, 8, 21, -14, 10], [-21, 28, 16, 12, -9, -18, 22], [-14, 9, -23, 2, 15, 9, 17], [-32, 24, 13, 34, -7, -19, 25], [-16, -17, 2, -13, -2, 26, 0], [-12, 6, 1, -8, -7, -16, 11], [34, 27, 10, 15, 7, -1, -6], [-1, -20, 3, -26, -4, 14, -2], [-26, -16, 3, -12, 0, 19, -13], [-10, 9, 3, 5, 12, -6, -17], [20, 20, 10, 12, 8, 7, 0], [25, -4, 27, 14, -31, -21, -12], [-10, 9, 0, 7, -13, -6, 12], [-23, -9, 1, -10, 3, 18, 0], [1, 26, 2, 27, 8, -9, 11], [-25, -10, -4, -18, -3, 12, -27], [-11, 11, 20, -2, -7, -19, 11], [-2, -7, -21, -26, -2, 2, -13], [-44, 22, 12, 25, 5, -22, 16], [18, -11, -20, -11, 18, 25, -12], [21, 5, 10, -11, 4, -8, -2], [-10, 10, -3, 12, -19, -5, 12], [-32, 20, 10, 19, -11, -36, -7], [-11, 14, -23, 0, 1, 7, 1], [20, -11, 17, -2, -2, 7, 7], [0, -7, 7, 5, -5, 1, -9], [-13, 20, 7, 4, 14, 9, 14], [11, 9, -3, 0, 3, -5, -14], [32, -21, -30, -19, 20, 25, -24], [-7, -13, 8, -9, -28, 2, -10], [17, 15, -12, -10, -10, 4, 15], [20, -9, 8, -14, -26, 22, -20], [11, -5, -5, 2, -1, 2, -12], [5, 1, 12, -6, 2, 25, -23], [-8, -19, 7, -9, -7, 21, -7], [12, -16, -9, 6, -2, 16, -2], [-2, -9, 18, 6, -19, 7, 8], [43, -6, 3, 4, 12, 13, -1], [17, -7, 19, -13, -9, 0, 2], [8, -15, -3, -3, -8, -4, -8], [21, -6, 4, 3, 16, 9, -17], [-19, -9, 22, 4, 4, 9, -20], [-27, -8, 10, -24, -14, 16, -3], [-11, -16, -4, -7, -11, -1, 16], [2, -15, 0, -27, 8, 22, -6], [0, 4, 6, -5, -1, 10, -32], [0, -13, 9, -6, -17, -3, -22], [-26, 13, -6, 13, -9, 3, 20], [0, -23, 9, -11, -10, -8, -7], [23, -3, -9, -1, 7, 2, -7], [3, -15, 5, -3, -3, -11, -13], [23, 19, 5, -4, -3, -12, -5], [2, 2, 9, 8, 5, 21, -2], [8, -13, 14, -24, -2, 4, -2], [-21, 7, -23, -16, 5, -3, -16], [-13, -5, -2, -3, -16, -2, -17], [35, -18, -8, -10, -17, 6, -7], [8, -5, 1, -2, 2, 10, -7], [-31, -13, 2, -22, 2, 25, -19], [-12, 0, 3, 22, 16, -21, 24], [12, -16, 6, 9, 10, -7, 3], [-24, 14, -1, 20, -10, -13, 13], [9, -20, -24, -23, 17, 33, 11], [-7, -5, -10, 18, 5, -12, 0], [29, -12, -4, 8, -2, -13, -6], [-39, 2, -8, 8, 3, 3, 19], [1, 11, 9, 1, -6, -9, -17], [19, -8, 4, 19, 13, 1, 15], [8, 14, 1, -2, 1, 11, -9], [-3, -19, -6, -7, 17, 23, -16], [-16, 5, 11, 24, 1, -8, -7], [6, 12, 2, 7, 27, 4, 6], [13, 9, 2, 31, 8, -14, 13], [12, 9, 32, 6, -4, 8, 1], [23, 8, -12, -3, 22, -18, 13], [-12, -2, -16, -5, 15, -3, 7], [-16, -6, -13, 10, -9, -14, 1], [13, 19, -6, 7, -7, -3, -7], [12, -2, 5, 6, -1, -20, -6], [-20, 3, 9, 20, 13, -3, 21], [27, -22, -26, -16, 4, 31, -15], [-14, -13, 6, -27, -24, 3, -10], [34, 4, -6, 9, 18, -17, -7], [5, 0, 26, 12, -19, -22, 13], [-4, 13, 21, -4, 12, -9, 2], [16, 2, 13, -1, 2, 3, 25], [4, 15, -21, -10, 14, -4, 20], [-7, 15, 17, 10, -21, -42, -4], [-33, 12, 14, 30, -16, -11, 13], [-19, 15, -6, 14, 1, -2, 25], [42, -10, 0, -3, 8, -4, 6], [21, 1, -13, -5, 13, -9, -4], [4, -9, 17, -5, -9, 10, -26], [-22, -29, 16, -6, -28, 16, -8], [4, 26, 12, 11, 9, 11, 9], [-48, 22, 23, 16, 12, -18, 14], [-30, 5, -4, -11, 10, 11, -5], [26, -13, -13, -5, 9, 8, -6], [-16, -6, 0, 15, 11, -10, -4], [8, 0, -5, -12, -10, 5, -15], [24, 3, 9, 9, 24, 12, 11], [13, 29, 5, 22, 0, -20, 10], [5, -14, -7, -11, -18, 11, -16], [21, 13, 18, 10, -9, -8, 13], [7, 19, -11, -25, -8, -5, -12], [5, -2, -9, -9, 31, 7, -18], [6, 16, 14, 39, -14, -32, 3], [8, -21, -1, -4, -19, -7, 2], [17, 32, -4, 9, 15, -8, -9], [20, 13, 27, 1, -2, 7, 6], [-5, 9, 8, 12, -18, 0, 6], [29, 17, 14, 1, -3, -13, 6], [11, 5, 5, 3, -12, -16, 16], [-9, -17, 1, 10, -12, -14, 22], [17, 10, 11, 16, -5, -3, 4], [5, 6, 2, -19, -12, -5, -13], [3, 8, -8, 10, 9, -23, -11], [-13, 3, -20, -8, 7, -1, -14], [36, 4, -17, -5, 3, -17, 11], [23, 29, -2, 4, 2, -10, 14], [0, -7, -30, -26, 28, 28, -6], [-16, 35, 12, 17, 0, -8, 31], [20, -13, -33, -8, 25, 15, -17], [15, -6, 10, -14, -10, 34, 2], [-9, 21, 10, 19, -1, -18, 4], [-44, -37, -4, -11, 3, 32, -10], [-7, -41, -16, -30, 24, 50, -12], [-11, 1, 2, 6, 12, -25, 11], [-17, 21, 23, 36, -16, -24, 16], [3, 22, -3, 0, -7, -42, -1], [-11, 16, -14, 0, 1, -2, -15], [23, -5, 4, 2, -21, 1, 1], [8, -29, -10, -10, -1, 5, -22], [-34, 12, -18, 18, -2, -16, -2], [-4, 19, -9, -2, 19, -7, 29], [-1, -28, -11, -29, 22, 37, -16], [24, -8, -26, -13, 20, 20, -21], [-12, -31, -2, 8, 0, 15, -1], [-29, -38, 5, -6, -21, 35, -30], [26, -8, 31, -10, -17, -7, -13], [17, 28, -15, -6, -15, -18, 17], [-23, 19, 5, 20, -4, -19, 6], [16, 19, -7, 8, 6, -14, 21], [-6, 26, 4, 3, 35, -10, 25], [27, 7, 18, 12, -20, -8, 7], [-43, 10, 11, 12, -17, -5, -12], [-15, -1, -7, 12, 17, -1, 8], [-9, 1, 4, -8, -2, 16, 1], [0, -19, 6, 5, 5, -21, 7], [-15, -29, -16, -2, 8, 4, -7], [-26, 12, 3, 3, -11, -7, -16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2667_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2667_2_a_j();".
// 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_j(:prec:=7)
    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_j();".
// 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_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2667_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![2, -11, 9, 14, -11, -7, 2, 1]>,<5,R![-5, -46, -138, -149, -42, 13, 8, 1]>],Snew);
    return Vf;
end function;