
// Make newform 9248.2.a.bq in Magma, downloaded from the LMFDB on 23 June 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_9248_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_9248_2_a_bq();".
// 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_9248_2_a_bq();".
// 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 := [37, 45, -87, -110, 48, 69, -7, -15, 0, 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, 1, 0, 0, 0, 0, 0, 0, 0], [-3, -1, 1, 0, 0, 0, 0, 0, 0], [19, -13, -43, 19, 22, -8, -3, 1, 0], [0, 19, -13, -43, 19, 22, -8, -3, 1], [17, 54, -68, -122, 75, 60, -27, -8, 3], [65, 42, -166, -88, 121, 44, -33, -6, 3], [98, 27, -250, -83, 176, 48, -46, -7, 4], [111, 51, -290, -121, 213, 64, -57, -9, 5]];
    Rf_basisdens := [1, 1, 1, 2, 2, 2, 2, 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_9248_a();"
function MakeCharacter_9248_a()
    N := 9248;
    order := 1;
    char_gens := [8671, 1157, 8673];
    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_9248_a_Hecke( : Kf := Kf)
    return MakeCharacter_9248_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], [1, -1, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 0, 0, 0, 1, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 1, 1, 0, 1, 0, 0, 0], [-2, 0, 0, -1, 0, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 1, 0, 1, 0, -2, -1, 1], [-1, -1, 0, 0, -1, 0, -1, 0, -1], [-3, 1, 1, -2, -1, 1, -3, 1, 0], [2, -2, -1, -2, -1, 1, 2, 1, -2], [-1, 2, 0, 1, 1, -1, 1, -3, 2], [3, -1, 0, 0, 1, 0, 0, 1, 2], [0, 0, 2, 2, 2, -1, 2, 1, -1], [-2, 2, 1, -3, 0, 0, -4, 0, -1], [1, -3, 0, 1, -1, 0, 0, 1, 0], [4, 2, 1, 2, 0, 2, -3, 0, 0], [-1, 0, 0, -1, 2, 2, -1, 0, -2], [-1, -1, -1, 0, 0, 3, 1, -2, -1], [-1, -1, 0, -1, -3, 1, -4, 3, 1], [4, -1, 1, -2, 1, -1, 1, 1, 0], [-1, 1, 1, 3, 2, -1, 4, 1, 0], [-1, -1, -1, -3, -1, 2, -5, 2, 0], [-2, -3, -2, 0, -5, 2, -2, 1, -2], [2, 2, 0, 2, -1, -1, 1, -1, -4], [-3, -1, -1, 0, 1, 1, 1, -3, -2], [-5, 1, 1, -1, 0, -2, -2, 0, 1], [-1, 3, -2, -3, -1, 1, 2, -1, -1], [1, -3, 0, 3, 2, -3, 7, 0, 0], [3, 3, 5, 4, 1, -3, -1, -1, 2], [3, 5, 2, 2, 5, 0, -3, -3, 5], [4, 0, 1, 0, 3, -4, -4, 2, 1], [-1, -4, 0, -4, -2, 0, 1, 2, -4], [0, 0, -3, -5, 2, 2, -2, -2, -1], [1, 1, 0, 1, 3, 0, -1, 1, 0], [0, 2, 2, -6, -1, -3, -7, 0, 0], [3, -1, -2, -2, -3, 2, 1, 3, -4], [7, -3, -2, 5, -2, 0, 1, 1, -5], [8, 4, -1, -1, 0, -4, -2, 1, -1], [-3, -4, 0, -3, 0, 0, 4, 4, 0], [-5, -1, -1, 1, 2, -2, 4, -1, 1], [-1, 2, 2, 4, 1, -3, -1, -1, 2], [1, 5, 3, 0, 2, 3, 3, -1, 1], [5, 5, 1, 2, 4, 0, -2, -2, 4], [4, 3, -3, -3, 4, 0, -1, 0, 2], [6, 2, 0, -5, 2, 0, -11, -2, 1], [4, 6, -2, -7, -2, 5, -3, 0, 0], [0, -2, 3, 2, 1, -3, -4, -2, 2], [8, 2, 2, 5, 4, -5, 7, -1, 2], [-7, 1, 2, -4, -1, 2, -9, -3, 0], [6, 5, 0, 3, 1, -2, -1, -5, 0], [-6, 4, 0, 0, 1, 3, 3, -2, -6], [4, 5, 7, 3, 3, -1, -7, -3, 4], [0, 4, 2, -5, 1, -2, 2, -4, 6], [-4, 0, 0, 7, -1, 5, 2, 3, -2], [9, 1, -6, 0, 0, 4, 0, 2, 0], [-12, -1, 0, 5, -1, 0, 2, -5, -4], [-12, 2, 1, 2, 0, -2, 3, 1, 0], [2, 5, 3, -7, 1, -1, -11, 1, 2], [2, 3, 3, -4, 4, -4, 1, -2, 6], [4, -10, -4, 3, -4, 2, 13, 2, -3], [-1, 3, 1, 6, 0, -2, 5, -2, -2], [8, -4, 2, 3, 3, -5, 4, 0, 5], [0, -10, -1, -6, 0, -5, 7, 0, -3], [0, 5, -3, 0, 2, 4, -5, 2, 0], [-3, -6, 1, -2, 0, 3, 2, 4, 2], [9, -1, 0, 5, -2, -1, -5, 2, 2], [-2, 3, 3, -9, 0, 2, -11, 0, 4], [8, 4, -2, -11, -2, 1, -9, -3, -2], [-1, -3, 3, 8, 3, -5, 2, -1, -2], [-9, 2, 4, -10, 0, 8, -5, 4, 0], [-1, -1, 0, -8, 4, -2, -6, 2, 8], [14, 2, 4, 3, 7, -5, -2, 4, 7], [-9, 4, 4, -4, 1, 5, -2, 3, 0], [-2, 6, 6, -5, -1, 3, -4, 0, 9], [-7, -5, 2, 8, -10, 2, 2, 4, -4], [5, -6, -2, -2, -7, 3, 3, 1, -10], [-2, -5, 2, 8, -2, -1, 5, 2, 4], [3, -6, -4, 0, -5, 1, 7, 1, -8], [-2, 5, 0, 8, 4, 1, 12, -8, 2], [8, -4, -5, 7, -5, 6, 9, 4, -8], [-1, 5, 2, 9, -1, 0, -4, -1, -2], [7, 3, 5, 9, 5, -5, 7, -5, 1], [-1, 5, 0, 3, -5, 0, 4, -1, 4], [7, 3, 0, 4, -1, 6, 7, 0, -1], [-4, 4, -5, 0, 0, 5, 7, -3, -3], [3, 6, -1, -1, 0, -1, -6, 0, 2], [0, 9, 2, 8, 4, 3, 8, -8, 2], [8, 5, 1, -1, 6, -4, 4, -2, 4], [3, -1, 0, 1, -4, -2, -3, 5, -9], [11, 3, -4, -7, -3, 7, -4, -3, 1], [-4, -6, -6, 2, -5, 3, -1, -1, 0], [5, -1, 6, 1, -1, -2, 4, 0, 0], [-9, 3, 1, -2, 2, -1, -9, -1, 3], [1, -3, -6, 1, -2, 2, 1, -2, -9], [6, -6, 3, 2, -6, 7, -3, 6, -3], [-6, -1, 3, -2, -3, -3, 9, -3, 0], [18, 4, -3, 3, 2, -1, 0, -6, 2], [-4, -4, 0, -3, -7, 4, 6, 5, 0], [-3, 4, 0, 3, 9, -1, 10, -5, 2], [10, 8, 6, 4, 4, -3, -2, -8, 5], [-1, 0, 0, 14, -4, -4, 17, -2, 0], [-2, 0, -4, -8, 3, 4, -3, -2, 1], [4, 9, 0, 9, -1, 2, 8, 1, -2], [1, 7, 6, 7, 5, 3, -2, -1, 3], [-2, 18, 8, 1, 3, -1, -15, -5, 0], [-7, -1, 2, 8, -3, -9, 1, 4, -6], [-4, -1, -3, -2, -5, 5, -9, 9, -10], [-3, -1, -9, 12, -6, 4, 11, -3, -4], [4, -3, 0, -10, 2, -1, -1, -6, 2], [15, 5, 0, 7, -5, 1, -6, 2, 1], [0, -8, -7, -13, -3, 0, -8, 3, -2], [11, 12, 1, 11, 1, -4, 8, -7, 2], [12, 4, 0, -2, -2, 0, -8, 2, -4], [4, 2, 0, 5, 5, 4, 4, 0, 2], [11, 4, 7, 4, 6, -9, -5, 2, 10], [15, -1, 2, 3, 4, -7, 11, 0, 2], [5, 9, 3, 6, 6, -3, 5, -7, 9], [-17, 7, 2, 2, 1, 2, 8, 1, -4], [2, -2, -4, -2, 0, -2, -6, 1, -10], [7, 6, 4, -11, -2, 2, -13, 0, 6], [-2, 4, 2, -8, -3, 3, -6, -3, 4], [-22, -3, -5, 7, -1, 3, 5, 3, -6], [-1, -11, 3, -4, -6, 2, -11, 6, -4], [-4, 4, 0, -3, 0, -6, 5, -5, -3], [1, 2, -2, -9, 1, -5, 2, -1, 8], [6, 3, 0, 4, 9, -8, 5, -9, 0], [3, 13, 3, -9, 4, 2, -4, -2, 1], [4, -6, 2, -5, 2, 2, -3, 5, 7], [-1, 2, -6, 8, 6, 2, 11, -4, -4], [-3, -1, 2, 14, 1, -4, 1, 1, -7], [1, 7, 9, 4, 11, -5, -4, -1, 6], [4, 2, 0, -13, 4, 5, -5, 5, 4], [-8, 13, 0, 15, 7, -6, 5, -11, 0], [2, 5, -6, -8, -4, 3, -13, 4, 2], [-6, 11, 6, -2, 7, -4, -17, -3, 6], [-13, 0, 5, 0, -5, -4, -4, 9, 2], [14, -2, -8, 7, -2, 5, 5, -1, -2], [-11, 8, 6, -3, 3, -1, -9, -1, 6], [18, 2, -2, -8, -7, -1, -8, 3, -4], [6, -4, 4, -11, -1, 0, -4, 5, 0], [-5, -11, -10, -10, -10, 1, -4, 6, -6], [15, 5, 5, 4, 2, -2, 1, 1, 8], [4, 8, -3, 3, 9, 3, -5, 0, 7], [-12, -1, 2, -8, 1, 2, 0, -1, -2], [-15, -1, 10, 0, 2, -1, 0, 1, -1], [-20, -9, 2, 2, -10, -1, -7, 0, -12], [8, 13, 4, -5, 7, 4, -15, -7, 2], [-18, 18, 7, 8, -1, 1, -2, -5, 4], [2, 8, 6, 7, 4, -3, -3, -2, 14], [-11, -2, 2, 6, 2, -10, -1, 4, -4], [5, 2, 4, -13, 0, 0, -14, -6, 6], [3, 11, 3, 0, 3, 2, -2, -6, 9], [5, -1, -4, -6, 1, -6, 1, 6, 7], [2, -2, 6, -5, -4, -1, -19, 0, 6], [-12, -2, 2, -4, 9, -7, 3, -3, -2], [-8, -24, -8, 0, -10, 1, 2, 4, -11], [2, 3, 1, -1, -2, 0, 13, -2, -2], [-5, -5, 2, 3, -7, -6, 9, 5, 2], [-12, 0, -6, 10, 3, -7, 9, -1, -2], [-16, -8, -11, 3, -2, 7, 12, 5, -10], [-8, 12, 4, 8, 6, -2, 5, -2, 2], [-1, -17, -5, -16, -9, 8, 2, 9, -7], [0, 8, 2, 14, 1, -2, -1, 4, 3], [18, 3, 3, 6, 6, -10, -3, -2, 2], [-9, -7, -9, 1, -2, 6, 6, -1, -9], [-11, 3, -6, 13, -6, 6, 1, 4, -11], [-8, 6, 1, 10, 5, 2, -6, 1, 6], [-14, 0, -3, 0, 1, 14, 0, 3, -4], [-6, -3, 4, -4, -5, 6, -7, -7, -6], [-10, 20, 5, -2, 3, 7, -10, -2, 8], [8, 6, 7, 4, 0, -3, -13, 4, 10], [-12, -4, -7, 2, -10, 11, -1, 10, -15], [13, -6, 3, -9, 1, 4, -7, 1, 6], [11, 9, -1, 7, -6, -2, 10, -7, 3], [1, -1, 0, 4, 0, 5, -6, 4, -12], [-4, 10, -7, 2, 2, 2, 13, -8, 0], [-19, -2, -2, -4, 1, 3, 3, -3, -2], [-2, -2, -3, 6, 6, -11, 5, -12, -1], [-8, -1, -4, -2, -13, 12, -14, 3, -4], [11, -9, -10, -17, 1, 1, -8, 4, -3], [-17, -11, -6, -6, -16, 2, -6, 7, -10], [8, 13, 0, -10, -1, 8, -13, -7, 8], [-14, 4, 8, 15, -2, -8, 4, -2, -2], [-14, 6, 1, 12, 6, 2, 1, -8, 8], [-4, -7, 0, 23, -10, 1, 10, 2, -10], [-11, -1, 5, 7, -11, -3, -1, 5, -8], [-11, -1, -2, 3, 2, 5, -1, -2, 4], [5, 3, 3, 4, 7, -7, -3, -5, 16], [-26, 12, 9, 3, -4, 0, -2, -3, 5], [-15, -1, -8, -13, -5, 8, -10, 1, -4], [-17, -3, -4, -3, -1, 15, 2, -3, -5], [-2, -8, -6, -4, -10, 1, 0, 5, -19], [-13, -10, -8, -25, -5, 5, 0, 1, -10], [-18, 8, 3, -1, 1, 4, -3, 1, -2], [22, 9, 2, 15, 2, -7, 2, 2, 8], [-1, -6, -8, 12, -9, 1, 20, 1, -8], [7, -13, 2, 1, 0, -5, 5, 8, -4], [-13, 2, -4, 3, 10, 2, 16, -6, 2], [5, -13, -8, -13, -1, 0, 4, 0, 2], [-2, 6, 10, 7, 9, -9, -8, -3, 16], [5, 3, 14, 12, 0, -2, 4, -3, 6], [10, 5, 5, 13, 2, -2, 1, 12, -2], [-4, -13, 1, 5, -12, 6, -1, 6, -14], [2, 18, 9, -9, 7, 7, -1, -2, 7], [-4, -3, 12, 16, -4, -3, 3, 0, 6], [17, -11, -5, -2, 4, -12, -3, -2, 6], [11, 7, 10, -3, 8, -7, 3, 3, 14], [24, -7, 1, -4, -6, 6, -6, 6, 4], [14, 6, 4, -8, 3, -16, 3, -6, 3], [-14, 4, 12, 12, 2, -4, 3, 0, -4], [-20, -16, -13, 5, -9, 6, 2, 5, -10], [21, -7, 1, -17, -7, -6, -9, 5, 2], [-3, 9, -9, 13, 3, -5, 17, 1, -5], [4, -4, -4, 1, 1, 2, 6, 3, 2], [15, 2, -5, -13, -5, 8, 3, 13, -2], [-10, 12, 7, -3, -3, 2, -21, -5, 6], [-10, -3, -5, 9, -13, 7, 9, 3, -8], [-22, -2, 0, 18, -5, 7, 3, 0, -6], [-19, -1, 1, 9, 7, -11, -1, -5, -6], [11, 0, -3, 10, -6, 9, 17, -4, -8], [16, 4, -2, 15, 0, -2, 7, -1, 3], [-7, 2, 6, 1, -6, 0, -24, -4, 10], [6, -10, -6, -20, -9, 4, 1, 5, -5], [-12, -14, 1, -1, -17, 6, -3, 17, -10], [-7, 6, 5, -8, -2, 3, -7, 2, 8], [3, -4, -2, -13, -6, 0, -10, 6, 0], [7, -1, -6, -2, -2, 3, -4, -1, 11], [12, -10, -3, 8, -5, 1, 28, 3, -10], [-30, 0, -1, 7, -7, 4, 1, -1, -4], [7, 2, -1, 0, 5, 8, -9, -1, -6], [-5, 7, 11, 9, 8, -8, 6, 4, -3], [10, -4, 4, -27, -5, 0, -14, 6, 0], [-2, -9, -4, -11, 6, -13, -3, -4, 2], [-7, 11, -5, -6, 7, 5, -12, -5, 0], [-17, -3, 8, -4, -7, 12, 3, 0, -1], [-21, -15, -8, -18, -12, 3, -16, 6, -16], [2, -9, 6, 12, 1, 2, 4, 5, -6], [-6, 8, 4, -4, 3, -4, 1, 2, 5], [8, -4, 0, 23, 4, -8, 9, -7, 5], [-15, -7, 5, -4, -7, 6, 4, 2, 3], [4, 8, -13, -8, 6, -1, -3, -3, 3], [9, -7, -10, 18, 10, 2, 38, -3, -2], [8, 7, 8, 15, 14, -1, -3, -2, 12], [-7, 17, 1, 21, 8, 0, -2, -10, 6], [-27, -3, -8, 5, -8, 0, 15, -6, -9], [27, -13, -2, 24, 5, -9, 35, -5, -6], [-12, 2, -1, 26, 3, -10, 16, -8, -3], [3, 7, 7, -14, 3, -6, 4, 10, 9], [4, 6, -8, 0, -11, 3, 3, 8, -6], [-4, -3, -5, -11, 5, 7, 20, 1, -12], [1, 1, 2, -7, -1, 8, -16, 1, 14], [-5, -11, -4, -14, -1, 9, -9, 5, 2], [-5, 16, 11, -1, -3, 4, -28, -3, 6], [21, 3, 8, 2, 3, -4, -8, -3, 16], [-38, -8, -3, 3, -12, 4, 4, 3, -11], [-3, -20, -1, -5, -12, 5, 1, 6, -4], [24, -10, -3, -13, -6, -1, 2, 9, 2], [-1, 8, -1, -9, -8, 1, -26, 10, -8], [-7, 10, 8, -10, 14, 0, -10, 6, 18], [8, -2, 7, 12, 14, -5, 9, -3, -1], [8, 10, -7, -4, 7, -7, 2, -3, 2], [11, 1, 11, 2, 4, -8, 9, 10, 12], [-4, 9, 4, 2, 6, 9, -20, -8, 10], [17, -8, -2, -9, -7, 15, 5, -1, -4], [-1, 7, 8, -4, -4, -3, -20, 2, 17], [11, -8, 1, -29, 10, -5, -18, 2, 8], [-8, 1, 3, -1, -8, 2, -3, 0, -8], [17, -7, -14, -10, 0, 0, 4, 4, 4], [6, 0, 3, -6, 1, 2, -16, -3, 14], [-23, -4, -10, 5, -7, 7, 8, -1, -6], [15, -5, 4, 24, 1, -2, 11, 0, -3], [22, 1, -6, -4, 4, -1, 12, 0, -8], [-8, -12, -4, 7, 7, -10, 26, -13, 0], [8, 10, 7, -10, 4, -11, 0, 0, 18], [15, 5, 4, 3, 6, -1, 13, -18, -2], [1, 9, 10, -20, 7, -9, -3, -8, 16], [-1, 1, -1, 1, -6, 6, -10, -4, 12], [0, -12, -16, -13, -6, 2, 2, 0, 4], [36, -6, -3, -12, 4, -11, 1, 4, 5], [-7, 1, 0, 4, -3, -3, 1, 6, 12], [-1, -1, 5, -20, -1, -2, -12, 5, -7], [-9, -3, 4, -15, -2, 4, -27, 3, 11], [12, -4, -15, -1, -10, -3, 0, 6, -8], [-10, -2, -1, 37, -3, -5, 23, -5, -5], [6, 2, -8, 21, -8, -10, 29, 5, -13], [-7, 1, -10, 2, -2, -7, 2, -10, 4], [-8, -7, -9, 9, -4, 6, 1, -2, -2], [-2, -2, -5, -2, -4, 14, -11, 0, 2], [-18, 4, -13, 2, 2, 7, 7, -10, -8], [-17, -8, 8, -19, 10, -10, -21, 0, 6], [2, -14, 2, 10, -3, -9, 17, 6, -14], [9, -19, -7, 19, -1, -5, 23, -1, -10], [-20, -14, -4, 0, -4, -8, 4, 5, -6], [-26, 6, 1, -15, -1, 8, -23, -9, 10], [0, -2, -7, -21, -9, 2, -9, 3, -14], [-23, -11, 2, 18, -3, -6, 17, 7, 7], [-1, 2, 4, 2, 1, -3, 8, -11, 4], [-29, 9, 6, 18, -2, -2, 6, -9, -2], [8, 10, 8, -2, 4, 9, 8, -4, -5], [-5, -11, 7, 21, -2, -6, 26, -8, 2], [-18, -13, -8, 11, -4, -7, 13, 10, 0], [-8, 0, 5, -9, 12, 4, -6, -10, 21], [-12, 0, -13, -3, 1, -11, 9, -1, -5], [-14, 10, 7, 27, 5, 6, 3, -13, -4], [-16, -4, 3, -17, 4, -1, -3, 6, 14], [-4, 0, 1, -6, 9, -4, 12, 3, 19], [-8, 5, 4, 0, 6, -3, -21, 0, 6], [6, -18, -1, -10, -8, 4, -3, 6, 0], [-13, -5, 2, -14, -7, 2, -35, 3, -4], [-15, 15, 14, -11, 16, -1, -1, -4, 20], [1, -5, 13, 2, 1, 5, 0, 5, -2], [21, 7, 0, -11, 9, -4, 13, -3, -2], [-6, -22, -9, -3, -9, -7, -9, 7, -13], [2, 14, -6, 8, 6, 13, -10, -13, 7], [29, 12, -1, 15, 4, -15, 11, -8, -6], [-22, 0, -2, -8, -6, 2, 14, -8, -12], [3, -1, 9, 0, 15, -9, 4, -12, 16], [-1, -4, 1, 27, 1, -16, 1, -5, 8], [19, -8, -2, -9, -9, 3, 0, -1, 6], [15, -5, 9, 5, 9, -9, -19, -5, 13], [2, 18, 8, 27, -2, -2, -6, 0, 8], [11, 8, -3, 8, 1, -4, 20, 1, -10], [-2, 10, -10, -2, 5, 1, -21, -5, -8], [-3, 0, -2, 31, 0, -8, 13, 8, -4], [19, -4, 6, 2, 11, -5, 3, 1, 12], [6, 6, 5, 18, 10, -2, 17, 3, 4], [3, -3, 5, -10, -1, 0, -28, 4, 15], [-14, -6, -12, -20, 5, 7, -5, 0, 0], [18, 12, 9, 6, 10, 5, 12, -8, 0], [-7, 11, 4, 12, 7, 0, 32, -9, 0], [-15, 13, -9, -12, -2, 10, -4, -4, 0], [24, -12, -5, -17, -10, 0, 16, 5, -13], [14, 6, 4, -1, 8, -14, -3, 4, -5], [0, 10, 8, 2, 2, 9, 8, -4, -1], [-8, -10, 3, -24, -9, 5, 0, 19, -2], [21, 4, -5, -17, -7, -6, -15, 11, -10], [-4, -13, -8, -13, -1, -6, -11, 3, -14], [-7, 0, -7, -7, -5, 18, 14, 15, -14], [-10, 0, 4, 23, 5, -9, 12, 7, 7], [7, -16, -3, 0, -15, -2, 1, 17, -22], [-18, 17, -3, -31, -9, 11, -29, 3, 0], [-5, 30, 4, 22, 8, -8, 7, -18, 8], [-8, 26, 5, 1, 1, 10, -11, -11, 4], [-2, 15, -3, -10, 13, 3, -17, -15, 12], [-8, 26, 13, 18, 4, -3, -3, -16, 5], [11, -5, 2, 19, 5, -10, -4, 3, -8], [-20, -6, -2, -22, -5, -2, -25, 0, -5], [-21, -3, 3, 15, 6, 9, 6, -4, -10], [17, -12, -2, -6, 6, -14, -4, -12, 14], [1, 11, 15, 2, 6, -1, 11, -14, 9], [-13, -16, 1, 5, 4, -9, -5, 8, -12], [-13, 17, 7, 4, 9, 3, 17, -11, 14], [-2, 2, -3, 16, -4, -19, 19, -1, -3], [-16, -7, 9, 5, 6, -8, 2, 6, 0], [23, 21, 14, 2, 5, 10, -17, -5, 0], [5, -21, 5, 7, 10, -14, 10, 6, 7], [-3, -7, -7, -26, 9, -3, 8, -6, 10], [6, -10, -2, -37, 6, 8, -27, 4, 4], [34, 14, 11, 6, 3, -10, 0, -13, 11], [8, -2, -5, 0, -7, -4, 9, 7, 10], [31, -17, -13, 6, -18, 0, 9, 0, -14], [16, 20, 14, -16, 20, -9, -16, -1, 15], [-4, 0, 0, -6, 0, -16, -26, 2, -4], [38, 4, -6, -7, -1, 3, 18, 4, -9], [-14, -3, 4, 8, -1, -16, 9, 3, -4], [9, -7, 7, -17, 4, -20, 2, 0, 8], [17, 9, 4, 5, -8, 2, -29, -5, -5], [23, -11, 11, -5, -3, -7, -16, 5, 14], [6, -6, -1, 0, 5, 11, -16, 0, 8], [1, 3, 1, -4, -4, -5, -27, -5, 5], [0, -22, -4, 4, 2, -17, 28, 9, -11], [26, -3, -3, -12, -1, -11, -19, 7, -2], [-21, 15, 11, 1, 12, -12, -26, 2, -1], [7, 14, 20, -14, 0, 8, -35, -2, 8], [15, -3, 14, 25, 6, -20, 3, 5, 13], [-7, -3, -10, 13, 9, -1, 22, -4, -15], [19, 9, 1, 10, 17, -7, 0, -3, 6], [1, 18, -7, -4, -2, 5, -19, -14, -2], [-20, 33, 12, -5, 12, -9, -16, -6, 2], [-8, 19, -4, -18, -8, 11, -19, -12, -2], [9, -7, -7, -22, -2, 12, 6, -8, -20], [9, 19, 14, 2, 7, -10, -13, 4, 9], [-18, -2, 3, -16, -6, 11, 8, 14, -4], [8, -18, -8, -10, -26, 7, 8, 8, -15], [5, 1, 6, 3, 0, -5, -9, 1, 12], [-6, 14, -4, 23, -4, 4, 19, 10, -11], [-6, -6, 2, 40, -9, -9, 32, 1, -6], [8, 2, 9, 25, 15, -3, 23, -4, 17], [15, -31, -11, -26, -13, 8, 8, 13, -13], [-4, 2, 1, -23, -7, -6, -5, 9, 2], [-19, 13, -4, -11, 0, 1, -22, 4, -6], [10, -4, 3, 15, -2, 7, 0, 9, 8], [18, 2, -20, -1, -1, 3, 10, 6, -7], [2, 2, -8, 5, 12, 4, 25, -9, 5], [-1, 10, -6, 10, -21, 13, -2, 7, -6], [-17, -3, -4, -22, -12, 20, -22, 0, 4], [16, -6, 5, 26, 23, -22, 23, 3, 2], [-22, -6, 12, -36, -10, 13, -22, 14, -5], [-10, -17, 3, 14, 4, -4, 9, 2, 20], [9, 1, -4, 6, 15, -8, 19, -7, 8], [-6, -6, 3, 21, -6, 1, 33, 12, -4], [0, 14, 9, 1, 6, 18, 4, -6, -9], [9, -25, -14, -12, -21, 16, 12, 19, -12], [2, 3, -1, 12, 0, -24, 0, -2, 12], [3, 17, 14, -27, -6, 18, -33, -6, 15], [-18, 11, -8, -35, -15, 8, -33, 9, -8], [-26, -1, -9, -18, -18, 8, -5, 6, -10], [20, -10, -13, -33, -3, 3, -9, 17, 11], [42, -24, -12, -2, 0, -5, 4, 7, -5], [-4, -9, -1, -16, 2, 2, 14, 0, -4], [-15, 18, 11, 22, 13, -16, 2, 1, 0], [-17, -25, -8, -12, -3, 22, 1, 8, -5], [19, -1, -7, -23, 8, -1, 6, -7, 10], [-21, 20, 5, -26, -8, 25, -40, 2, 0], [-12, -2, -1, 34, -2, -15, 12, 10, -8], [9, -11, -3, 12, -21, 2, -2, 2, -5], [16, 6, -5, 17, -9, 16, 23, -3, -10], [-4, -13, 2, -23, -12, 1, 11, -2, -20], [-7, -7, -6, -22, 0, 6, -20, 4, 2], [5, -10, -11, 25, 9, 2, 38, -1, -8], [18, 27, 4, -12, 8, -13, -5, -2, 10], [10, -10, -4, 6, -5, -13, 39, 10, -14], [12, 2, -2, 15, 1, -6, 12, -1, -4], [-4, -21, -8, -13, 2, 1, 11, -10, -8], [16, 11, -7, -11, 1, 5, 0, -7, -18], [-22, 14, 8, -15, -10, 21, -15, 2, 4], [-10, -4, 4, -1, -6, 24, -7, 6, -8], [-29, -9, -4, 13, -9, 1, 24, 0, -11], [-19, -1, 8, -15, 13, 10, -24, -4, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9248_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9248_2_a_bq();".
// 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_9248_2_a_bq(:prec:=9)
    chi := MakeCharacter_9248_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_9248_2_a_bq();".
// 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_9248_2_a_bq( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9248_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![19, -60, -177, 191, 111, -162, 28, 21, -9, 1]>,<5,R![1, 51, -315, 324, 432, -33, -97, -9, 6, 1]>,<7,R![17, -48, -378, -521, -99, 177, 49, -24, -3, 1]>,<19,R![2447, 6663, 1731, -3862, -870, 873, 41, -57, 0, 1]>,<43,R![-8890633, 16074114, -4688835, -1553569, 144075, 38238, -1222, -339, 3, 1]>],Snew);
    return Vf;
end function;