
// Make newform 8700.2.a.bj in Magma, downloaded from the LMFDB on 06 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_8700_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_8700_2_a_bj();".
// 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_8700_2_a_bj();".
// 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 := [-6, -10, 11, 12, -7, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [-1, 2, 0, 0, 0, 0], [-1, -4, 0, 1, 0, 0], [2, 4, -5, -1, 1, 0], [7, 0, -7, 0, 1, 0], [-5, 4, 11, -6, -2, 1]];
    Rf_basisdens := [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_8700_a();"
function MakeCharacter_8700_a()
    N := 8700;
    order := 1;
    char_gens := [4351, 5801, 4177, 901];
    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_8700_a_Hecke( : Kf := Kf)
    return MakeCharacter_8700_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], [-1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [1, 0, -1, 0, 0, 0], [0, 0, 0, 0, 1, 1], [0, -1, 1, 0, 0, 0], [0, 0, 0, 1, 1, -1], [-2, -1, 0, 1, 0, -1], [1, 1, 1, 0, 1, 2], [1, 0, 0, 0, 0, 0], [-2, 0, -1, 0, 1, -2], [0, 0, -2, -1, 2, 0], [-2, -2, 1, -1, -1, 0], [6, -1, 1, -1, -1, -1], [0, -2, 3, 2, 0, 1], [4, 1, 0, 1, 0, 1], [0, 1, 0, 2, 0, 1], [-4, 1, -2, -1, 0, 1], [7, 0, 0, -1, -1, 0], [2, 1, -3, -2, 1, -1], [-2, 0, 0, 1, 0, 2], [6, -3, 2, -1, -2, -3], [3, 0, -1, -3, -1, -1], [-2, 0, 0, 0, 1, 3], [-2, -1, -3, 0, 1, 1], [-2, 0, 0, 0, -1, 1], [7, 3, -2, 2, 0, 0], [7, 3, -1, 0, 1, 0], [2, 3, -3, 1, 0, 0], [2, 0, 2, -1, -1, -3], [2, -2, 2, -2, -4, 2], [-4, 2, -2, 2, 5, 1], [-2, 0, -2, 0, 2, -2], [0, -1, -1, -1, 0, -2], [-6, 1, 0, 2, 0, 1], [0, 0, 1, 1, -1, -2], [4, 1, -1, 1, 3, 1], [8, 1, -3, 1, 1, -3], [-1, 1, -1, 1, 1, -2], [2, 0, -4, 0, 0, -2], [6, 1, 3, 0, -1, 3], [-2, -1, 4, 2, -1, 4], [-8, 0, 6, 2, 0, 4], [2, 4, -2, -2, 0, 0], [4, -4, 0, -1, -4, -4], [0, 1, 4, 0, 1, 4], [-4, 2, 4, 0, 0, 4], [5, -2, 1, 2, 0, 4], [3, 4, -1, 3, -1, 3], [8, -2, 1, 0, -3, -4], [4, 0, 4, 2, -2, 2], [-2, 0, 5, 0, 1, 2], [10, 1, 0, 2, 5, 4], [0, -2, 3, 5, 2, 1], [-8, -5, 6, 2, -2, 7], [4, -2, -4, -2, 2, 0], [-18, 0, 3, 3, 2, 1], [4, 0, -4, -2, -4, -2], [8, -1, 4, 3, -1, 4], [-2, -7, 4, -2, -2, -1], [-1, -5, -2, 2, 2, -2], [2, 0, -2, 1, 3, -3], [4, 0, 0, -5, 0, 0], [-4, -4, 1, 1, 2, -3], [10, 1, -1, -4, 0, -6], [4, 2, -1, -2, -2, -1], [6, 4, -3, 0, 5, 0], [-6, -2, 0, 1, 2, 2], [5, -3, 7, 2, -5, 6], [10, 0, 2, -4, -2, 0], [14, 2, 0, -1, 0, 0], [-8, 2, 1, -3, -3, -4], [8, -2, -4, -4, -4, -6], [10, 6, -5, -3, 5, -2], [14, 0, -5, -4, 3, 0], [5, 2, -3, -2, 7, 1], [2, 4, 1, -1, 2, -1], [-2, 0, 1, -3, -1, 0], [-4, 1, -3, 2, -3, -1], [2, -1, -1, -1, -1, -5], [6, -4, 5, -6, -7, -2], [2, 1, -1, 3, 7, 1], [-4, -4, 4, 2, 0, 8], [-4, 3, 1, 4, 1, 7], [0, -3, 2, 2, -1, -4], [12, 2, 3, 2, -2, 3], [2, 4, -3, -3, -2, 7], [-2, 1, 0, 5, 5, 2], [10, 0, 3, 1, -5, 8], [7, -2, -2, 5, 5, -2], [2, 0, 1, -1, -1, 8], [-12, 6, -7, -3, 1, 0], [11, 1, 4, 2, 0, 8], [16, 4, -4, -4, 2, -4], [8, -5, -3, -3, -4, 0], [4, -6, 1, 2, 0, -1], [-14, -1, 0, 6, 2, 5], [-14, 1, 6, 4, 2, -3], [15, 0, 0, -5, -5, 4], [12, 4, -1, 2, 3, -6], [-1, -2, 1, 2, -2, 0], [-6, -1, 0, -2, -2, -1], [10, 6, 0, -3, -1, 1], [10, 2, -4, 6, 1, 1], [8, 2, -5, 1, -5, 2], [-2, 2, -6, -4, -2, -8], [3, -2, -1, 3, -5, -7], [-2, 3, 4, 5, -4, -3], [4, -4, -4, 0, 1, -7], [0, -2, -1, -6, 3, -4], [-14, 1, 7, -1, 3, 7], [4, 1, 4, 3, 1, 8], [-4, 2, -4, 0, 2, -4], [14, 2, -5, -4, -5, -2], [-4, 1, 1, 1, 2, 4], [6, 2, 8, -2, -5, 3], [5, -4, 7, 0, -6, -2], [-1, 0, 3, 4, 7, 7], [14, -6, 5, 2, -4, -7], [-8, -2, 2, 8, 1, 1], [10, 1, -10, 4, 3, -4], [-10, 3, 2, -5, -5, 0], [6, -4, -4, 3, 7, -1], [1, -1, 1, -5, -1, -6], [12, -1, -1, 1, -2, 8], [-20, -1, 1, -6, -5, 3], [-2, -4, -2, 6, 2, 0], [0, -5, 8, 0, -2, 1], [10, 0, -2, 3, -6, -6], [-6, 3, -1, -1, 1, -5], [2, -1, -2, -3, 0, -7], [-8, 2, 8, 5, -1, 3], [0, 4, -4, -6, -2, -8], [-4, 2, 8, 2, 8, 10], [16, 1, 5, -6, -3, -1], [0, -1, 8, -1, -6, 3], [-2, 0, 2, 6, 4, 0], [11, -11, 6, 0, 0, -2], [12, 2, 4, 0, -8, -2], [-14, 6, -7, 1, 2, -3], [12, -5, 3, 9, 0, -10], [-6, -2, 4, 8, 0, 2], [16, 3, -1, -1, 1, -7], [26, -4, -1, -1, -5, -10], [14, -4, 4, 6, 2, 2], [-12, -6, 11, 9, -4, 7], [18, -3, 5, -7, -1, -3], [2, 2, -14, -3, 2, -4], [16, -1, -3, 5, 1, -11], [1, 0, -3, 6, 5, -3], [-18, -2, 7, 3, -1, 4], [-10, 4, 3, -1, 0, -5], [1, -5, -2, 0, -8, -6], [-4, 4, -12, 1, 13, -1], [-2, 1, 1, -3, 9, 7], [12, 2, -1, -5, -4, 3], [16, -1, -2, -2, -3, 0], [-10, -6, -4, 2, 4, -2], [-14, -1, 6, -5, -9, 0], [-6, -1, -8, 2, 4, 3], [-12, -2, 0, 1, 5, -3], [-12, 5, -12, -2, 0, 1], [10, 5, 1, -2, 9, 7], [-4, -2, 2, 4, 6, 8], [-10, -2, 6, -1, 4, 6], [-6, -4, 1, -3, -7, 2], [-8, 1, 2, -6, 7, 8], [-12, 0, 0, 4, 0, -8], [26, -3, -2, 0, -1, -4], [-12, 0, 2, 5, 1, -3], [-14, -6, 5, 0, -1, 2], [10, -1, 3, 1, -6, -4], [2, -2, 9, -2, -1, -6], [-18, -4, 6, 2, -2, -2], [16, 8, 2, -4, -2, -4], [-24, 7, -7, 4, 3, 7], [-4, 1, -7, -9, 12, 6], [-10, -4, 9, -9, -6, 3], [-6, -6, 0, -1, -10, 2], [10, 3, 10, -8, -5, 2], [14, 7, 1, -5, -1, 7], [12, 2, -4, -4, 7, -3], [-4, 1, 6, 3, 1, 2], [-2, -2, 6, -11, -8, -6], [9, -2, 1, 8, -3, -1], [12, 3, -5, -12, 5, -1], [8, 1, 9, 4, -5, 7], [-20, 4, 4, -1, 6, 10], [-2, 7, -13, 1, 9, -7], [-4, 5, -8, 2, 0, 1], [2, 1, 1, -6, 6, 6], [-16, -2, 5, 8, 2, 9], [-22, 6, -5, 0, -3, 0], [-10, 4, 11, 1, 2, 9], [-14, -10, -7, 2, 4, -11], [-14, 5, 4, -3, -2, 1], [-2, 8, -12, 2, 10, 2], [0, -1, -7, -10, 4, -4], [-8, 5, -4, -6, -2, -1], [1, -1, 13, 2, -1, 6], [-10, 8, -14, -6, 8, 2], [0, -1, -7, -1, 0, 0], [-20, 6, -6, 3, 4, 4], [8, -7, 6, -3, -10, -1], [-4, -4, 0, -4, -7, 5], [2, -7, 14, 4, -4, 1], [0, -8, 4, -6, 0, 2], [-5, 4, 7, -2, -7, -7], [0, -1, 7, 10, 5, 3], [-6, 12, -3, 4, 7, -2], [2, 1, -7, -5, 1, 11], [-8, 0, 1, 4, -11, -6], [12, 8, 4, 4, -8, 4], [2, -2, 3, -2, -8, 11], [0, 0, 6, -8, -1, -3], [-6, 1, 12, 4, -1, 10], [-9, -8, 13, 10, -4, 8], [-10, -2, -5, 7, -2, -5], [19, -3, -3, 2, 3, 2], [2, -2, 2, 5, -1, 3], [14, -9, 5, -9, -13, -11], [0, -11, 0, 0, -1, -6], [10, 4, 1, 5, 4, -7], [-26, 9, -9, -6, 1, 5], [-1, -5, 11, 3, -9, -2], [24, 2, -5, -2, -5, 0], [-14, 0, -1, 3, 3, 18], [4, 10, -4, -2, 8, 8], [10, -2, 8, 6, -12, 0], [-6, -2, -1, -2, -7, -14], [-10, 1, -15, -1, 11, -3], [20, 0, 1, 1, 7, 6], [-2, 3, 4, 1, -6, -3], [-14, -2, 3, 7, -9, -2], [3, -8, -5, 2, -2, -16], [-14, 8, -1, 0, 4, 13], [0, 1, 6, 9, 0, 1], [20, 0, -15, -1, 7, -6], [-2, -2, 3, -4, -7, 6], [-4, 10, -4, 10, 5, 3], [-28, 6, -4, 11, 9, 5], [8, -8, 2, 2, -4, -6], [1, -10, 6, -3, -1, -2], [-2, 0, -2, 4, -4, -6], [-14, 3, 10, 3, 2, 9], [-18, 4, -11, -4, 11, -4], [19, -7, 11, 5, -15, -10], [18, 6, 1, -12, 1, 0], [12, -3, -6, -6, 1, 10], [2, 4, 5, -2, -6, 1], [-8, 1, 5, -2, -15, -3], [-24, 0, -3, 2, 9, -2], [18, -2, -2, -3, 5, 1], [14, 3, -7, -9, 5, -1], [-2, -14, 10, -8, -16, -2], [-24, -1, -6, 0, 4, 1], [-2, 9, 5, 3, -7, -3], [-8, 2, 6, -6, -8, 4], [6, 6, 0, -2, -6, 2], [2, 1, -10, -7, 3, -2], [14, 6, -2, -13, -2, 0], [14, 6, -6, 7, -5, 3], [34, 3, 3, -5, -2, 0], [-14, 3, 3, 13, 9, 7], [-12, 4, 9, -1, -5, 2], [4, 1, -5, 9, 5, 1], [-30, -4, -7, 3, -4, -7], [-10, 2, -15, -5, 15, 0], [-34, 4, 10, 10, -2, 2], [-30, 5, 0, 2, 10, 9], [6, -9, -1, -3, 6, 4], [13, 6, -9, 2, 8, -6], [12, 8, -9, 3, 3, 0], [2, 6, 1, -6, -3, 6], [20, 3, -7, -8, -3, 3], [-8, 4, -12, 4, 2, -6], [-4, 2, 8, 6, 6, 12], [22, -3, 11, 5, -2, -2], [-42, 3, 11, 15, 7, 3], [24, -6, -12, -6, -4, -14], [-17, 6, -9, 2, 5, 11], [-4, 5, -15, 1, 0, -6], [15, 1, 1, -3, -7, -8], [2, 1, -9, 3, 8, -8], [-16, 6, 4, -1, -2, -8], [-20, 0, 1, -3, -8, 7], [-14, 12, -6, -5, -6, -6], [-20, -4, 2, 9, 11, -1], [28, 9, 2, 0, -5, 8], [-6, 4, -2, -4, 4, 2], [4, 5, 3, -6, 1, -7], [-15, 11, 3, 2, 1, 6], [-20, 2, 6, 6, 2, 2], [0, 0, 9, -3, -1, 16], [-6, -3, -1, -12, -3, 1], [-8, -6, 15, 0, -1, 12], [-28, -7, 5, -9, -1, -3], [14, 4, -16, 2, 8, -14], [26, 1, 1, 8, -11, -9], [-4, -1, 3, 4, 3, 19], [-8, 4, -10, -6, -10, -8], [-36, -1, -2, 1, -4, -7], [-12, -9, -7, 7, 3, -1], [38, 2, -6, -10, 0, -10], [-20, 0, -8, -6, -6, -16], [8, 2, -14, -12, 12, 0], [0, -2, 1, 0, 4, -13], [-2, 12, -14, -10, 2, -4], [22, 5, -15, -4, -1, -19], [-12, -13, 18, 5, -3, 4], [-13, 12, -1, 6, 9, 9], [-10, 6, 7, 11, 8, 17], [8, 6, -11, -4, -1, 2], [5, 2, -7, 8, 8, 8], [21, 0, 3, -7, -3, -17], [6, -3, -3, -7, 2, 8], [0, 8, 9, -3, 1, 2], [42, -6, 7, 4, -7, -2], [-34, -3, 5, 4, 6, -6], [34, 8, -5, 7, 0, -9], [0, 13, -1, 3, -5, -5], [-26, -9, -5, 2, 0, 0], [-16, -15, 1, 6, -5, -11], [14, -7, 7, 10, 5, -7], [-4, -7, -2, 6, -2, 9], [-6, -17, 7, -3, -5, 3], [-16, 8, -10, 6, 16, 10], [-7, 9, -4, 0, 14, 16], [0, 0, -1, -4, 0, -9], [12, 0, -4, -15, 4, -12], [14, -6, 0, -4, 6, 4], [16, 9, -8, 1, 2, -5], [36, 4, -8, -12, -4, -4], [0, 10, -16, 1, 1, 1], [0, 5, 5, 3, -3, 7], [-25, -11, -1, 11, 7, -4], [22, -1, -2, -2, -5, -18], [-36, -16, -1, 4, -4, -9], [6, 6, -8, -14, -4, 10], [-35, 4, -12, -1, 15, -2], [-10, -5, 1, -10, -1, 3], [12, -4, -4, 0, 0, -4], [30, 4, 1, -7, -12, 9], [-8, -11, 2, -2, -5, -8], [14, 10, 4, 0, 8, 12], [-16, 0, 11, 11, -1, 6], [26, -5, 11, -1, -15, -3], [4, 3, -13, -8, 15, 3], [-16, 2, -6, 8, 5, 1], [-26, 4, -16, 3, 3, -1], [12, 0, 6, -8, 2, -8], [6, 7, -15, -2, 4, -14], [22, 4, -10, -12, 8, -6], [9, 1, 5, -11, -3, 10], [-14, -4, -6, 14, 14, -12], [0, 10, -6, -2, 6, 2], [4, -4, 7, -9, 2, -9], [-36, -8, 4, 4, -4, -8], [8, -12, -2, -3, 9, 1], [21, -2, 15, 3, -17, -1], [0, -13, 3, 0, 2, -4], [4, 12, -9, -8, 3, -6], [10, 4, -6, -7, -7, 15], [-24, 6, 2, 0, 4, 12], [-54, -7, 3, 2, -3, 1], [-28, 1, 15, 18, -5, 9], [52, -6, -2, 1, -5, -15], [-9, -8, 9, 16, 2, 6], [28, -5, 6, -15, -10, -11], [-36, -8, 11, -5, 5, 0], [-16, -1, 1, 5, 6, 2], [-7, -2, 5, 10, 5, 11], [30, -6, 11, 3, -12, -7], [-44, -7, 10, 4, -5, 8], [2, -2, 9, -3, -1, -6], [8, 4, 4, 10, -17, -1], [-4, 2, 1, 15, 4, 1], [4, -3, -7, 7, 9, -7], [-8, -3, 11, 6, -3, 13], [4, 6, 7, 9, -3, 4], [-14, -6, -5, 1, 1, 12], [-6, -2, -6, 1, -12, -18], [-18, -12, 6, 7, 8, -12], [-28, -2, -17, 6, 8, -7], [12, 10, -10, -12, 2, -2], [-5, 0, 1, -7, 9, -1], [22, 12, -21, 2, 5, -10], [8, -1, 10, -11, -19, 4], [20, -10, 14, -10, -8, -12], [-32, 14, -6, 3, 1, -1], [24, 11, 4, 7, -6, 15], [-2, -21, 16, -4, -20, 1], [24, 8, -14, -12, 5, -17], [30, -4, 12, -1, -8, 10], [24, 6, 5, -7, -2, -17], [6, 3, 7, 0, -6, 6], [-40, 2, 16, -2, -2, 12], [-6, 8, 20, 0, -4, 12], [-24, -7, 10, 4, 3, 0], [2, 0, 8, 2, -7, -9], [2, -2, 0, 6, -8, 2], [-30, 10, -9, 1, 1, 6], [2, 7, -19, -8, 5, -23], [0, 4, 6, -6, -6, -8], [-26, -7, 0, 4, 0, -19], [-20, -6, 4, 0, 8, 6], [12, 1, -3, -8, 8, -6], [18, -4, -7, -9, -2, -21], [40, 0, 4, -11, -4, -12], [10, 8, -1, -2, 3, 10], [-22, -7, 4, -9, -11, -14], [54, 9, 4, -6, 6, 1], [17, -2, -11, -8, -3, -5], [-4, 8, 10, 8, 6, 16], [20, -14, 12, 3, -14, -12], [10, 2, -5, -5, 6, 11], [-20, 0, -2, 2, -15, -5], [35, 2, -2, 3, 7, 12], [22, -10, 20, 7, -1, 1], [-14, 6, -8, -2, 10, -6], [16, -11, 5, -8, -13, 1], [-8, 1, 9, 7, 13, 13], [-26, -8, -2, 0, 0, 0], [28, -10, 4, -8, -4, -8], [22, 1, -30, -5, 7, -14], [10, -15, 22, 4, -14, 3], [9, -11, 21, 13, -7, 10], [-6, -6, -7, -3, -8, -1], [-32, 3, -1, 3, 2, 8], [16, 0, -1, 11, 5, 12], [-26, -1, 9, 9, -7, 7], [32, 0, -2, 12, 6, 0], [-16, 11, -11, -9, 1, -7], [-7, 9, -9, -1, -13, -2], [-30, 11, -1, 6, 1, 3], [-38, -6, -2, -2, -6, -2], [-46, -7, 2, 10, -3, 2], [10, 3, -7, -23, -8, -12], [-21, 0, 8, 1, -1, -4], [12, 11, -2, -4, 3, -2], [3, 13, 15, -1, -1, 24], [-46, -14, 6, 0, 1, 7], [6, -3, 14, 10, 7, 16], [36, 5, 4, 2, 10, 7], [34, 7, -17, -3, -7, -7], [-46, 12, -6, 5, 8, 0], [14, 10, -20, -12, 14, -6], [-13, 8, -15, -1, -11, -5], [-24, 0, -7, 2, 7, 4], [-14, 4, 4, 4, 2, 6], [22, 6, -12, 1, 12, -8], [18, 1, -13, 4, -1, -9], [-23, 0, -9, -9, 9, 11], [-10, -4, 6, -2, -13, -1], [-52, 4, -6, -8, 0, 14], [-34, -6, -3, -5, 6, -11], [4, 8, -3, 2, -17, -8], [4, -6, -5, 13, 10, -3], [34, -4, -3, -1, -9, -20], [22, 5, 0, -1, -16, 7], [-50, -2, 3, -12, -5, 6], [-8, 3, -17, 5, 12, 6], [12, 0, -10, 6, 14, 8], [18, 8, 0, 4, 22, 2], [-1, -8, 14, 13, 1, 4], [6, 11, -7, 10, 16, 10], [-32, -5, -5, 5, 4, -14], [32, -8, -3, -16, -10, -1], [12, 8, -17, 8, 7, -14], [4, 0, -5, -5, 13, -2], [2, -4, 14, 7, 12, 12], [46, -4, -1, -3, 5, 8], [-18, -13, 3, -6, 5, -13], [16, -7, 8, 1, -16, -17], [28, -4, 6, 14, -10, -8], [6, 6, -19, 5, 13, -12], [-40, -3, 3, -2, 3, -9], [-42, -1, 6, -5, -12, 5], [0, 6, -7, -11, -5, -16], [18, 11, -10, 14, 8, 7], [-42, -7, -1, -5, -3, 1], [-2, -2, 1, -7, -5, -8], [-30, -7, 3, 2, 4, -14], [2, -6, 18, -4, 5, -5], [16, -3, -3, -4, -9, -13], [51, -5, 13, 9, -7, -6], [2, -4, -14, -6, 0, -14], [-12, -6, 1, 3, 8, -3], [-36, 2, 14, -6, -8, -2], [-6, -21, 8, 3, 4, 3], [-28, -6, 4, 2, -8, -2], [-16, 8, -6, -7, -17, -5], [-2, -11, 7, 9, 4, 4], [22, -11, 24, 4, 0, -1], [10, 7, -5, -2, 1, 7], [20, 3, -8, 5, 14, -9], [15, 0, 7, -4, -14, 12], [8, -2, -2, 2, 0, -16], [-2, 2, -27, -4, 9, -2], [-12, 5, 15, -7, -22, -2], [6, -8, -8, 4, -3, 3], [5, 4, -18, 5, 15, -8], [14, -6, 18, -7, 3, -9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8700_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8700_2_a_bj();".
// 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_8700_2_a_bj(:prec:=6)
    chi := MakeCharacter_8700_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(3593) - 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_8700_2_a_bj();".
// 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_8700_2_a_bj( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8700_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, -48, 13, 74, -10, -6, 1]>,<11,R![172, -236, -89, 160, -32, -4, 1]>,<13,R![144, -544, 209, 78, -34, -2, 1]>],Snew);
    return Vf;
end function;