
// Make newform 8700.2.b.l in Magma, downloaded from the LMFDB on 22 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_8700_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_8700_b_Hecke();"
// 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_b_l();".
// 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_b_l();".
// 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 := [8, -8, 6, -6, 3, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 4, -1, 0, -1, 0], [4, 0, 5, -1, 1, -1], [2, -2, 1, -2, 1, 0], [-8, 2, -4, 3, 0, 1], [-20, 8, -11, 8, -3, 4]];
    Rf_basisdens := [1, 2, 2, 1, 2, 2];
    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_b();"
function MakeCharacter_8700_b()
    N := 8700;
    order := 2;
    char_gens := [4351, 5801, 4177, 901];
    v := [2, 2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_8700_b_Hecke();"
function MakeCharacter_8700_b_Hecke( : Kf := false)
    N := 8700;
    order := 2;
    char_gens := [4351, 5801, 4177, 901];
    char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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], [0, -1, 0, 0, -1, 0], [0, -1, 0, 0, 1, 0], [0, 0, 0, 0, 1, -1], [0, 0, -1, 0, 0, 0], [0, 1, 0, 0, 1, 1], [0, -1, 0, 0, -1, -1], [1, 1, 1, 1, 0, 0], [0, -1, 0, 0, -1, 1], [0, 0, -2, 0, 0, 0], [0, -4, 0, 0, -2, 2], [2, 0, 0, 1, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, -1, 2], [-2, 0, -2, -1, 0, 0], [0, -2, 0, 0, -2, 2], [0, -1, 0, 0, 1, 0], [0, 0, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, -3, 0, 0, -3, -1], [0, 3, 0, 0, 1, 1], [0, 2, 0, 0, 2, -1], [4, 0, 0, -2, 0, 0], [0, -4, 0, 0, 0, 1], [0, -1, 0, 0, 5, -1], [0, -5, 0, 0, -3, 1], [4, 0, 3, 1, 0, 0], [8, 0, -1, 0, 0, 0], [-8, 0, 2, 2, 0, 0], [0, -3, 0, 0, -1, 4], [-2, 0, 0, -1, 0, 0], [-8, 0, -1, 2, 0, 0], [6, 0, -2, 0, 0, 0], [-10, 0, -2, -1, 0, 0], [8, 0, 2, 0, 0, 0], [-6, 0, -2, 1, 0, 0], [0, -1, 0, 0, -1, 5], [0, 0, 0, 0, -1, 0], [-4, 0, 0, -4, 0, 0], [0, 0, 1, -3, 0, 0], [0, -1, 0, 0, -1, -1], [-8, 0, 2, 0, 0, 0], [0, -4, 0, 0, 3, 2], [-10, 0, 7, 1, 0, 0], [0, 3, 0, 0, -1, 5], [0, 5, 0, 0, -3, -2], [0, -3, 0, 0, 3, 3], [0, -2, 0, 0, -6, 2], [0, -2, 0, 0, -7, 0], [6, 0, 2, -3, 0, 0], [-4, 0, -3, 1, 0, 0], [0, -9, 0, 0, -7, -4], [0, -6, 0, 0, 1, 0], [-16, 0, 0, 2, 0, 0], [0, -4, 0, 0, 0, 1], [0, -3, 0, 0, -3, -3], [0, 0, 0, 0, -5, 1], [-2, 0, 4, 2, 0, 0], [0, 5, 0, 0, 1, 1], [-8, 0, -5, 0, 0, 0], [0, 0, 2, 2, 0, 0], [0, 3, 0, 0, 13, 0], [0, 6, 0, 0, 1, -1], [8, 0, 3, 4, 0, 0], [0, 1, 0, 0, -3, 5], [0, 0, 2, 4, 0, 0], [0, -5, 0, 0, -7, 1], [10, 0, -8, -2, 0, 0], [0, -2, 0, 0, 7, 4], [0, -3, 0, 0, -3, 5], [10, 0, 4, 5, 0, 0], [0, 6, 0, 0, 3, -6], [0, -9, 0, 0, -1, 5], [0, 1, 0, 0, -3, 3], [0, -4, 0, 0, -8, -1], [0, 6, 0, 0, 5, -6], [14, 0, 0, 2, 0, 0], [0, 8, 0, 0, 8, -2], [-14, 0, 0, -3, 0, 0], [0, -6, 0, 0, -2, -6], [10, 0, 4, -1, 0, 0], [8, 0, -2, 0, 0, 0], [-18, 0, 1, 1, 0, 0], [14, 0, -5, 1, 0, 0], [0, 6, 0, 0, -2, 1], [0, 2, 0, 0, 3, 7], [0, -6, 0, 0, -4, 4], [0, 9, 0, 0, 5, 4], [8, 0, 0, 2, 0, 0], [0, -13, 0, 0, -1, 3], [0, 9, 0, 0, 7, -3], [0, -5, 0, 0, -13, -1], [24, 0, 1, -2, 0, 0], [6, 0, 5, 1, 0, 0], [-22, 0, -2, -2, 0, 0], [2, 0, 4, -2, 0, 0], [0, 7, 0, 0, 5, 2], [0, -2, 0, 0, -10, 8], [0, 5, 0, 0, 7, -2], [0, 0, 0, 0, 7, 0], [20, 0, -3, 2, 0, 0], [0, 6, 0, 0, 6, -3], [-8, 0, -4, -4, 0, 0], [4, 0, -10, 2, 0, 0], [0, 1, 0, 0, 3, -3], [0, -6, 0, 0, 7, -2], [0, 5, 0, 0, 11, 4], [0, -4, 0, 0, 4, 0], [22, 0, 0, -1, 0, 0], [0, -4, 0, 0, -13, -1], [18, 0, -4, 1, 0, 0], [0, -9, 0, 0, -1, 5], [6, 0, -5, 1, 0, 0], [0, -2, 0, 0, -10, 9], [0, -5, 0, 0, 1, -6], [0, 5, 0, 0, -3, 5], [-16, 0, -3, -4, 0, 0], [0, -3, 0, 0, -17, 2], [4, 0, -11, -3, 0, 0], [0, 18, 0, 0, 3, -7], [-4, 0, 9, -2, 0, 0], [0, -3, 0, 0, 7, -3], [-8, 0, 3, -6, 0, 0], [-26, 0, 4, 2, 0, 0], [-2, 0, -4, -8, 0, 0], [-6, 0, 4, -1, 0, 0], [-6, 0, -2, -7, 0, 0], [12, 0, -10, -2, 0, 0], [0, 5, 0, 0, 1, -5], [-14, 0, 13, 3, 0, 0], [0, -1, 0, 0, -5, -5], [-8, 0, 6, 4, 0, 0], [-2, 0, 4, 0, 0, 0], [0, 16, 0, 0, 4, -6], [-6, 0, 4, -3, 0, 0], [0, 3, 0, 0, 19, -1], [-10, 0, -8, 3, 0, 0], [0, 14, 0, 0, -2, -5], [-22, 0, -5, -3, 0, 0], [-34, 0, 0, 2, 0, 0], [20, 0, 6, 8, 0, 0], [18, 0, 2, -7, 0, 0], [0, -2, 0, 0, -2, 4], [0, 1, 0, 0, 11, -6], [16, 0, -6, 0, 0, 0], [0, 6, 0, 0, 9, -4], [0, 7, 0, 0, 11, 3], [0, -3, 0, 0, -3, 7], [0, -6, 0, 0, 3, -6], [0, -2, 0, 0, 6, 1], [0, 3, 0, 0, -1, 7], [2, 0, -5, 3, 0, 0], [-2, 0, 2, -5, 0, 0], [0, -1, 0, 0, 13, -6], [22, 0, -1, 5, 0, 0], [10, 0, 2, 8, 0, 0], [0, 10, 0, 0, -1, -7], [2, 0, -4, -6, 0, 0], [-2, 0, -11, 1, 0, 0], [0, -18, 0, 0, -3, 10], [24, 0, -4, -2, 0, 0], [0, 3, 0, 0, 3, 3], [0, -2, 0, 0, -7, 2], [-24, 0, 0, 6, 0, 0], [-4, 0, -1, -6, 0, 0], [-36, 0, -4, 2, 0, 0], [4, 0, -11, 5, 0, 0], [-20, 0, 1, -6, 0, 0], [-18, 0, -6, -5, 0, 0], [-8, 0, -1, 1, 0, 0], [20, 0, -4, 2, 0, 0], [-44, 0, 8, 2, 0, 0], [32, 0, 4, 2, 0, 0], [10, 0, 0, 4, 0, 0], [36, 0, -1, 0, 0, 0], [0, 4, 0, 0, 24, -7], [0, 0, 2, 10, 0, 0], [-4, 0, 1, -7, 0, 0], [-28, 0, -2, 4, 0, 0], [0, -15, 0, 0, -5, -4], [0, 8, 0, 0, -13, -3], [0, 6, 0, 0, 1, 0], [0, -9, 0, 0, -5, 3], [2, 0, 2, 12, 0, 0], [48, 0, 2, 4, 0, 0], [26, 0, 0, 5, 0, 0], [0, 8, 0, 0, -16, -2], [6, 0, 0, -3, 0, 0], [0, 2, 0, 0, -1, -5], [24, 0, 1, 0, 0, 0], [0, 5, 0, 0, 17, -3], [0, 0, 0, 0, -4, -5], [18, 0, -3, -9, 0, 0], [0, -6, 0, 0, -7, 10], [0, 12, 0, 0, 16, -2], [0, 20, 0, 0, 3, -3], [0, 10, 0, 0, 3, 0], [0, -11, 0, 0, -3, 3], [0, 2, 0, 0, -8, 4], [22, 0, -3, 5, 0, 0], [16, 0, 10, 12, 0, 0], [0, -4, 0, 0, 0, -8], [0, -3, 0, 0, -25, 2], [0, 16, 0, 0, 15, 8], [0, -17, 0, 0, -13, 5], [0, 5, 0, 0, 11, -11], [-10, 0, 14, 2, 0, 0], [0, 7, 0, 0, -17, -1], [12, 0, 6, -10, 0, 0], [22, 0, -8, -2, 0, 0], [-26, 0, -2, 7, 0, 0], [8, 0, -7, 4, 0, 0], [0, -7, 0, 0, -13, -6], [-28, 0, 1, 3, 0, 0], [0, -1, 0, 0, 15, -6], [0, 6, 0, 0, 6, -13], [0, -19, 0, 0, -15, -3], [0, 0, -7, -12, 0, 0], [0, 18, 0, 0, 10, -8], [-16, 0, 7, 4, 0, 0], [0, 2, 0, 0, 10, -11], [-40, 0, 0, -6, 0, 0], [0, 7, 0, 0, 9, 3], [0, -6, 0, 0, 26, 2], [14, 0, 8, -9, 0, 0], [0, -1, 0, 0, -5, 7], [-18, 0, -4, -5, 0, 0], [-12, 0, 18, 8, 0, 0], [20, 0, 4, -2, 0, 0], [-4, 0, 4, -6, 0, 0], [0, 9, 0, 0, 5, -9], [0, 8, 0, 0, -14, -6], [0, -3, 0, 0, -17, -6], [18, 0, 5, -9, 0, 0], [0, -8, 0, 0, -16, -6], [10, 0, -8, 1, 0, 0], [36, 0, -10, -4, 0, 0], [0, 11, 0, 0, 13, 0], [-38, 0, -7, 7, 0, 0], [22, 0, 2, -7, 0, 0], [0, -5, 0, 0, 11, -12], [0, -6, 0, 0, -26, 0], [0, -2, 0, 0, -3, 8], [36, 0, -10, -2, 0, 0], [0, -11, 0, 0, -1, 9], [8, 0, -8, 6, 0, 0], [-34, 0, 3, 3, 0, 0], [22, 0, -9, -3, 0, 0], [12, 0, 8, -6, 0, 0], [6, 0, -8, 8, 0, 0], [-2, 0, -5, -11, 0, 0], [0, 8, 0, 0, -12, 4], [-10, 0, -8, -1, 0, 0], [-44, 0, 10, 0, 0, 0], [0, 2, 0, 0, -10, -10], [-26, 0, 0, -5, 0, 0], [0, -20, 0, 0, -13, 6], [0, 22, 0, 0, 9, -5], [18, 0, 4, 5, 0, 0], [-12, 0, 5, -2, 0, 0], [8, 0, -9, 7, 0, 0], [32, 0, -8, -4, 0, 0], [26, 0, -16, -7, 0, 0], [0, 7, 0, 0, -5, 11], [0, 0, 0, 0, -4, 1], [0, 20, 0, 0, 2, -6], [12, 0, -6, -8, 0, 0], [0, 8, 0, 0, 9, -2], [8, 0, -11, 3, 0, 0], [0, 7, 0, 0, 5, 0], [0, 12, 0, 0, 15, -12], [0, 5, 0, 0, 5, 7], [16, 0, -18, -8, 0, 0], [24, 0, 4, -6, 0, 0], [12, 0, 8, 8, 0, 0], [-16, 0, -11, 3, 0, 0], [0, 16, 0, 0, -8, 0], [24, 0, 10, -2, 0, 0], [0, -3, 0, 0, 9, 9], [-22, 0, -1, -5, 0, 0], [0, 1, 0, 0, 5, -3], [20, 0, 1, -7, 0, 0], [-28, 0, -10, 0, 0, 0], [0, 7, 0, 0, -11, 2], [28, 0, -10, -2, 0, 0], [32, 0, -1, 0, 0, 0], [-4, 0, 9, 2, 0, 0], [38, 0, -2, 4, 0, 0], [2, 0, 10, 2, 0, 0], [0, -5, 0, 0, -3, -9], [0, -6, 0, 0, -19, -4], [0, -7, 0, 0, 13, -3], [24, 0, -16, -4, 0, 0], [38, 0, -10, -4, 0, 0], [0, 7, 0, 0, -9, -11], [0, 8, 0, 0, -1, 2], [38, 0, 4, -1, 0, 0], [50, 0, -8, -3, 0, 0], [-8, 0, 12, 0, 0, 0], [0, -12, 0, 0, 1, 10], [0, 11, 0, 0, 3, 11], [-18, 0, 2, 7, 0, 0], [0, 25, 0, 0, 17, -9], [0, 16, 0, 0, -9, -8], [-24, 0, -7, 4, 0, 0], [50, 0, 4, 2, 0, 0], [-42, 0, 2, -3, 0, 0], [0, 4, 0, 0, 19, 3], [0, -13, 0, 0, -5, 11], [0, -8, 0, 0, 12, -13], [46, 0, -4, 2, 0, 0], [0, -11, 0, 0, -9, 4], [0, -23, 0, 0, -15, 5], [-8, 0, 7, 5, 0, 0], [0, 21, 0, 0, 13, -7], [26, 0, -4, 7, 0, 0], [0, 10, 0, 0, 13, -19], [0, -10, 0, 0, -26, -3], [0, -13, 0, 0, -1, -7], [0, 22, 0, 0, 27, 1], [18, 0, 10, 2, 0, 0], [-50, 0, 2, 3, 0, 0], [0, -2, 0, 0, 21, -6], [0, 16, 0, 0, -4, -4], [-6, 0, 2, 11, 0, 0], [0, -13, 0, 0, -1, -9], [-14, 0, -5, 3, 0, 0], [0, 4, 0, 0, 5, -10], [0, -2, 0, 0, 26, -2], [0, -8, 0, 0, -27, 10], [30, 0, 6, -1, 0, 0], [10, 0, 11, 3, 0, 0], [0, -15, 0, 0, -11, 1], [0, 11, 0, 0, 5, 9], [-36, 0, -7, 1, 0, 0], [80, 0, 3, 0, 0, 0], [0, 0, 0, 0, 0, -2], [0, 15, 0, 0, 15, 10], [-60, 0, 8, -2, 0, 0], [0, 10, 0, 0, 11, -2], [0, 24, 0, 0, 4, -3], [18, 0, 7, 11, 0, 0], [0, 4, 0, 0, 23, -2], [0, -3, 0, 0, 29, -7], [0, 10, 0, 0, 2, 8], [70, 0, 0, 3, 0, 0], [0, -23, 0, 0, -13, 0], [60, 0, 3, -2, 0, 0], [-10, 0, 6, -7, 0, 0], [0, 22, 0, 0, -9, -5], [0, -2, 0, 0, 8, -16], [0, 11, 0, 0, -15, -9], [0, 2, 0, 0, -10, -8], [-6, 0, -4, 5, 0, 0], [0, -1, 0, 0, -11, 0], [-30, 0, -2, 1, 0, 0], [-16, 0, -3, -8, 0, 0], [0, -23, 0, 0, -23, -11], [0, 8, 0, 0, 9, -15], [-26, 0, -6, 14, 0, 0], [-36, 0, -5, -2, 0, 0], [32, 0, -6, -10, 0, 0], [-14, 0, -6, 1, 0, 0], [-20, 0, -2, -10, 0, 0], [8, 0, -9, -4, 0, 0], [0, -15, 0, 0, -19, 12], [0, 28, 0, 0, 28, 0], [0, -9, 0, 0, -9, -1], [44, 0, 11, -4, 0, 0], [0, 15, 0, 0, 7, -9], [0, -4, 0, 0, -24, -1], [-28, 0, -5, -2, 0, 0], [0, 6, 0, 0, 13, -22], [0, 5, 0, 0, -9, 14], [0, -7, 0, 0, -5, -8], [16, 0, 14, 16, 0, 0], [18, 0, -4, 0, 0, 0], [0, 4, 0, 0, 7, 12], [0, -10, 0, 0, 4, 4], [0, 6, 0, 0, 7, -10], [-40, 0, 0, -10, 0, 0], [52, 0, -9, -2, 0, 0], [-4, 0, -8, 2, 0, 0], [0, -9, 0, 0, -5, -1], [0, 9, 0, 0, 25, -7], [0, 24, 0, 0, 19, -3], [-46, 0, -10, -3, 0, 0], [34, 0, 7, 11, 0, 0], [0, 8, 0, 0, -16, 6], [0, 36, 0, 0, 11, -10], [0, -3, 0, 0, -21, 4], [36, 0, 4, -4, 0, 0], [0, -15, 0, 0, -1, -2], [0, -6, 0, 0, 17, 13], [-22, 0, 2, 5, 0, 0], [0, 24, 0, 0, -2, -14], [28, 0, -5, 8, 0, 0], [0, -12, 0, 0, 0, 19], [10, 0, 0, -14, 0, 0], [-14, 0, 4, 5, 0, 0], [0, 0, 14, -2, 0, 0], [0, 2, 0, 0, -25, 8], [26, 0, -20, 2, 0, 0], [40, 0, -12, -6, 0, 0], [0, 12, 0, 0, 29, -13], [0, 22, 0, 0, 22, 13], [-18, 0, 4, 11, 0, 0], [-16, 0, -16, -6, 0, 0], [0, -6, 0, 0, -31, -1], [0, 0, 0, 0, -19, 12], [0, -9, 0, 0, -15, 1], [14, 0, 2, 1, 0, 0], [24, 0, -12, 0, 0, 0], [0, -8, 0, 0, 4, 5], [0, 9, 0, 0, 19, -2], [0, -23, 0, 0, -3, 8], [-28, 0, -9, -14, 0, 0], [18, 0, 10, 13, 0, 0], [2, 0, 16, -8, 0, 0], [-8, 0, 4, 12, 0, 0], [2, 0, -6, 1, 0, 0], [0, -7, 0, 0, 25, -3], [0, -18, 0, 0, -5, 5], [0, -16, 0, 0, -21, -10], [0, 15, 0, 0, -3, 5], [0, -6, 0, 0, 2, 5], [60, 0, 1, 0, 0, 0], [0, -11, 0, 0, -35, 9], [0, -4, 0, 0, 4, -14], [-22, 0, 10, 5, 0, 0], [0, -3, 0, 0, 17, 11], [0, -7, 0, 0, 25, 3], [-32, 0, 2, -4, 0, 0], [22, 0, -12, -14, 0, 0], [-24, 0, 3, -3, 0, 0], [-24, 0, -11, -13, 0, 0], [0, -17, 0, 0, 17, 6], [2, 0, -5, -5, 0, 0], [0, 29, 0, 0, -13, -11], [0, 6, 0, 0, 6, 3], [-12, 0, 1, -15, 0, 0], [-30, 0, 4, -5, 0, 0], [0, 4, 0, 0, -24, 8], [0, 18, 0, 0, 35, -4], [-18, 0, 2, -1, 0, 0], [0, -9, 0, 0, 15, -11], [0, 0, 0, 0, 16, 0], [-34, 0, 12, 2, 0, 0], [24, 0, 16, 18, 0, 0], [-42, 0, -4, -7, 0, 0], [0, 11, 0, 0, 25, -19], [0, 18, 0, 0, 26, 3], [-4, 0, 6, 6, 0, 0], [0, -4, 0, 0, -24, 11], [0, 14, 0, 0, 6, 16], [0, -7, 0, 0, 19, 4], [0, -14, 0, 0, -25, -6], [0, -10, 0, 0, -21, 28], [0, -17, 0, 0, -1, 9], [20, 0, 7, 2, 0, 0], [-28, 0, -6, -12, 0, 0], [-18, 0, -16, 2, 0, 0], [0, -11, 0, 0, -1, 20], [0, -14, 0, 0, -27, 21], [-12, 0, -13, 10, 0, 0], [44, 0, -7, -2, 0, 0], [14, 0, 4, 14, 0, 0], [2, 0, 6, 15, 0, 0], [2, 0, 18, -7, 0, 0], [-22, 0, 6, 5, 0, 0], [2, 0, 10, 3, 0, 0], [0, 12, 0, 0, -4, -6], [-38, 0, -4, 13, 0, 0], [0, -6, 0, 0, -23, 10], [46, 0, 12, 6, 0, 0], [0, 5, 0, 0, -19, 5], [-14, 0, 18, 9, 0, 0], [0, 16, 0, 0, 13, -24], [20, 0, -10, -10, 0, 0], [0, -20, 0, 0, -34, -10], [0, -2, 0, 0, 1, -19], [0, -20, 0, 0, -8, 15], [-40, 0, -10, -2, 0, 0], [0, 9, 0, 0, 39, -13], [0, -6, 0, 0, 6, 14], [0, 13, 0, 0, 39, -4], [0, -1, 0, 0, 19, -9], [0, -3, 0, 0, -39, 15], [40, 0, 4, 4, 0, 0], [24, 0, -9, -8, 0, 0], [-56, 0, 15, 3, 0, 0], [0, 0, 0, 0, 9, -8], [8, 0, -8, -14, 0, 0], [0, -22, 0, 0, -18, 8], [0, -13, 0, 0, -3, 6], [6, 0, 16, 3, 0, 0], [0, -35, 0, 0, -23, 13], [4, 0, 1, 4, 0, 0], [0, -20, 0, 0, -8, 11], [0, 9, 0, 0, -11, 16], [12, 0, -11, -10, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8700_b_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8700_2_b_l();".
// 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_b_l(:prec:=6)
    chi := MakeCharacter_8700_b();
    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_b_l();".
// 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_b_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8700_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![64, 0, 64, 0, 17, 0, 1]>,<11,R![64, 0, 256, 0, 41, 0, 1]>,<17,R![16, -16, 1, 1]>],Snew);
    return Vf;
end function;