
// Make newform 9405.2.a.bq in Magma, downloaded from the LMFDB on 07 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9405_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_9405_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_9405_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 := [1, -25, 142, 142, -567, -240, 766, 193, -467, -80, 137, 15, -19, -1, 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, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2295, -14395, -19347, 56138, 25300, -77197, -13260, 46919, 4271, -13308, -735, 1753, 48, -87], [2229, -23747, 272, 113948, -28564, -170354, 27141, 111088, -6073, -33204, 201, 4558, 31, -234], [5737, -57051, -76518, 208675, 164555, -265500, -136958, 154635, 51593, -44749, -8736, 6253, 538, -335], [-533, 32051, -51152, -173731, 143439, 299259, -114577, -214857, 30523, 67442, -2890, -9496, 61, 494], [-3281, 34088, 2269, -180017, 25177, 304017, -19419, -217238, -2536, 69009, 2252, -9920, -233, 529], [-7043, 66041, 47987, -296002, -91961, 442812, 87667, -291107, -43059, 89438, 8946, -12769, -633, 685], [3067, -50531, -6660, 254334, 599, -424595, -13611, 301535, 19327, -96018, -5683, 13894, 471, -747], [12592, -128747, -83587, 485923, 147381, -643200, -124027, 388805, 54428, -113373, -10619, 15632, 726, -818], [9178, -128178, -79604, 485923, 146812, -643200, -124027, 388805, 54428, -113373, -10619, 15632, 726, -818], [-15099, 183072, 108156, -681319, -200483, 899522, 176382, -544590, -79252, 159244, 15602, -22031, -1070, 1157]];
    Rf_basisdens := [1, 1, 1, 1, 569, 569, 569, 569, 569, 569, 569, 569, 569, 569];
    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_9405_a();"
function MakeCharacter_9405_a()
    N := 9405;
    order := 1;
    char_gens := [1046, 1882, 5986, 496];
    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_9405_a_Hecke(Kf)
    return MakeCharacter_9405_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, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0], [-1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -1, 1, 0, 0, -1, 1, -2, 0, 0, 0, 0, 0], [-1, 0, 2, -2, 1, -1, 0, -2, 2, -2, 0, 0, -1, -1], [0, 1, 1, -1, 0, 0, -1, 0, 0, -1, 0, 1, 1, 0], [-2, -1, -1, 0, 0, -1, 1, 0, 0, 0, -1, -1, 1, -1], [-1, 0, 2, -1, 0, -1, 0, 0, 1, -1, 0, 0, 0, 1], [-2, 0, -3, 2, -1, 0, 0, 1, -3, 3, 0, 0, 0, 1], [0, 1, 1, 1, 0, -1, -1, 0, -1, -1, -1, -1, 3, 1], [-2, 0, -4, 2, -3, 1, 3, 1, -1, 3, 0, -1, -1, -1], [1, -1, 0, 1, -2, 1, 0, 1, -2, 2, 1, 1, 0, 0], [1, 1, -3, 2, -1, 1, 0, 1, -1, 3, 3, 0, -1, 0], [0, 2, 0, 1, 0, 2, 1, 1, 1, 0, -1, 1, 0, 0], [1, 2, 1, 0, 1, 1, 1, -1, 0, 1, 1, 2, 0, 0], [0, 0, 2, -1, 1, 0, -1, 0, 2, -3, 1, 2, 0, 1], [0, 0, 5, -2, 2, 0, -1, -1, 2, -3, -1, 2, -1, 1], [1, 1, -3, 4, -2, 1, 1, 1, -2, 1, 0, 0, 0, 2], [1, 3, 4, -4, 2, 0, 1, -2, 4, -2, -1, 2, -2, -1], [-1, 2, -2, 1, -1, 3, 3, -1, -1, 2, 1, 1, -1, 1], [0, 2, -3, 2, -1, 2, -1, 3, -2, 3, 1, 0, 0, -1], [-1, 0, 4, -1, 2, -1, -3, -1, -1, -2, -1, 3, 0, 1], [-2, 2, -5, 4, -4, 3, 0, 1, -4, 3, 1, 0, 0, 1], [-1, -2, 0, -1, 0, 0, 3, 1, 2, 0, 0, -1, 1, -1], [-3, 2, 5, -5, 4, -2, 0, -2, 4, -5, -3, 1, 2, 0], [-2, 0, 0, 1, 0, 1, 3, -2, 1, 1, 0, 2, -3, 0], [-1, 3, -2, 2, -2, 3, 2, 1, -1, 1, 1, 1, 0, 0], [1, 1, -1, 1, 0, 1, 1, -2, 0, 3, 2, 1, -2, 0], [-1, -1, -3, 2, -1, 1, 2, 1, 1, 3, 3, 0, -3, 0], [1, 0, -1, 1, -1, -1, -2, 2, 1, 1, 2, -1, 1, 1], [-3, -1, 2, -2, -2, -1, 1, -2, 3, -2, -2, 0, -1, -1], [0, -3, -5, 5, -3, 0, 0, 3, -4, 4, 1, -2, 1, 1], [0, -3, -2, 1, -3, -1, 0, 3, -3, 2, 0, -3, 1, 0], [-4, -2, 0, 0, 0, -1, -2, 1, -2, 1, -1, -1, 3, 2], [-2, 1, 0, -3, -2, 1, 3, -1, 4, 0, -1, -1, -1, -1], [2, -3, -5, 3, -3, 0, -1, 4, -6, 3, 0, -3, 1, -1], [-2, -2, -2, 1, -1, 1, 2, 2, -4, 3, 1, -2, 0, 0], [5, -3, 11, -7, 4, -3, -4, -1, 6, -8, -3, 0, 2, -2], [-6, -5, 5, -4, 4, -4, -2, -2, 2, -6, -3, -1, 1, -2], [-1, 1, 1, 0, 0, -1, -3, 3, 0, -1, 0, -2, 2, 2], [-2, -3, -3, 3, -2, 0, -3, 2, -6, 3, 0, -1, 0, -1], [-8, 3, 0, -2, 2, -2, -4, 1, -1, 1, -1, -1, 4, -1], [-6, -2, 4, -2, 3, -2, -3, 1, 0, -3, 1, 1, 3, 0], [-4, 2, 0, -1, -1, -1, -2, 1, -1, 4, 1, -1, 1, -1], [-2, 1, -5, 1, -2, 3, 5, -2, 1, 1, -1, 1, -3, -1], [-4, -4, 4, -4, 4, -5, -4, -3, 0, -5, -1, 1, 0, -1], [-2, 0, 1, -1, 3, -1, -3, -1, -2, 0, 0, 2, -1, 3], [-2, 0, 9, -9, 3, -2, 4, -6, 9, -5, -4, 1, -2, -1], [2, -1, 1, 0, 1, -4, -3, 2, -1, -4, -2, -3, 4, 0], [-4, 0, 1, -1, -1, 2, 2, -2, 3, 1, 2, -1, -4, -1], [-5, 3, -2, -2, 1, 3, 6, -4, 4, 0, 1, 2, -5, -2], [-4, -1, -1, -2, -2, 2, 0, -3, -1, -3, -1, 2, -2, 1], [-3, 2, 3, -3, 3, 0, -1, -3, 4, -2, 2, 0, 2, 3], [-1, 5, -1, 3, -4, 3, 4, 2, -1, 3, -1, 1, -1, 1], [-1, -5, 9, -9, 8, -2, 0, -7, 7, -8, -2, 2, -1, -2], [-3, -1, 3, -1, -2, 3, 2, -2, 3, -1, 1, 1, 1, 1], [-4, 1, 2, 0, -1, 0, 1, -3, 0, -1, 0, 3, -3, 3], [-6, -5, -1, 2, -2, 0, -4, 4, -4, 2, 3, -1, 3, 0], [-1, 0, -2, -1, 1, -1, 1, 1, 1, -2, 1, -1, 2, 0], [0, 0, 2, 0, 2, -2, -6, -2, 0, -2, 2, 4, 0, 0], [1, 2, -3, 3, -4, 4, 2, 0, -2, 5, 1, 3, -4, 0], [-6, 4, -5, 2, -2, 0, -2, 4, -2, 4, 0, -3, 3, -1], [-2, -4, 0, 0, -2, 0, 2, 3, -3, -1, -4, -1, 3, -1], [-4, -8, 4, -4, 2, -5, -4, 1, 0, -5, -1, -3, 2, -3], [-5, -3, 9, -5, 5, -3, -5, -2, 0, -7, -4, 1, 2, -1], [-1, 2, -14, 8, -5, 2, 4, 4, -5, 10, 1, -4, -1, 0], [-2, 4, -8, 6, -3, -3, -5, 5, -7, 5, 2, -3, 4, 0], [-2, 1, 7, -3, 5, -4, -4, -3, 4, -8, -1, -2, 3, 1], [0, 4, 10, -5, 4, -3, -3, -2, 7, -5, 0, 0, 1, 2], [-10, -6, 0, -1, 3, -3, -2, 0, 0, -3, 1, -4, 4, 0], [-5, -4, 0, -2, 0, 1, -5, 3, -4, 1, 3, 1, 1, 2], [1, 3, -5, 3, -2, 7, 8, -3, 4, 4, 1, -2, -4, -2], [-3, 0, -2, 4, -2, 2, 3, 1, -3, 3, -2, -3, -2, 2], [5, -5, 2, 0, -1, -5, -5, 2, -3, 0, 2, -2, 4, 1], [-1, -2, 5, -6, 4, 1, 0, -4, 2, -6, 2, 1, 0, -1], [3, -3, -4, -2, 2, 1, -1, 0, -3, 0, 0, 0, -2, 0], [-10, -5, 2, -2, -1, 1, 3, 2, 0, 2, -1, -4, -2, -1], [10, 1, 1, 0, -4, 3, 6, -1, 4, 1, -4, 0, -3, 1], [-5, 13, 1, 2, 1, 2, 3, 0, 2, 2, -1, -1, -2, 2], [-2, 1, 1, 3, -2, -1, -5, 0, -3, -5, -1, 3, -1, 1], [1, -8, -4, 0, 1, 0, 2, 0, 1, -2, -3, -4, -1, 0], [7, 0, 3, 0, 0, -2, 3, -1, 7, -5, -2, -2, 3, -1], [2, -1, 0, 4, -2, 3, 0, 4, -1, 4, 2, 0, 0, 0], [5, 0, 5, 4, 4, -1, -5, 3, -2, 1, 3, 4, 4, 5], [-4, 2, -4, 2, -4, 6, 6, 0, 4, 4, 4, 2, -4, -2], [1, 1, 6, -6, 1, 3, 6, -5, 5, -1, -1, 3, -3, -4], [9, -3, 3, 1, 0, -1, 0, -1, 2, -4, -3, 0, 3, 3], [1, -1, 9, -5, 4, 0, -3, -4, 5, -3, 1, 5, -1, -2], [4, 0, 1, 3, -1, 5, 1, 1, -4, 6, 4, 4, -5, 1], [0, -2, -3, 5, -2, -1, 0, 0, -4, 1, 1, -3, -3, 2], [-5, 5, -1, 2, 2, 2, -1, 3, -1, 3, -1, 0, -1, 2], [6, -4, 2, -2, 2, 0, 8, 0, 6, 0, 0, 0, 1, 1], [9, 1, 2, 2, 1, -3, -5, -4, -1, 0, 2, 0, 2, 1], [-6, 0, 8, -4, 4, -4, 0, -1, 7, -5, -2, -1, 1, -3], [-6, 10, -4, 3, -2, 2, 3, 1, -1, 6, -1, -1, -3, 1], [-13, -4, -3, 3, 0, -1, 2, -1, 0, 4, 2, 0, -6, -3], [5, 5, -2, 2, 1, 3, 0, -2, 2, 6, 3, 0, -3, 0], [-10, -6, -4, 0, 0, -3, -5, -2, -2, -2, 0, -2, 0, -3], [-3, -2, 0, -1, 0, -1, -4, 2, 1, -5, 0, -2, 2, -3], [1, 5, -3, 2, 1, -1, -2, -3, -3, -3, 1, -2, -1, 0], [-3, 1, 5, -5, 1, 2, 1, -4, 3, -3, -1, -1, -1, -3], [-2, -1, 0, 5, -3, 1, 1, 0, -5, -1, -1, -2, 1, 2], [-1, 9, 0, 2, 3, 1, -4, -2, -2, 2, -1, 2, -1, -2], [-1, -4, -9, 5, -2, -1, 2, 2, -3, 3, 0, -3, -2, -3], [0, 5, 1, 1, 1, 4, -2, 1, -2, 2, 1, 4, -2, 0], [-1, 5, -2, -2, 2, 4, 5, -5, 3, 1, 3, 5, -2, 3], [4, 0, 0, -1, 5, 1, 6, -5, 3, 0, 1, 1, -4, -2], [1, 0, -2, 5, -1, -2, 3, 1, 0, 4, 0, 1, -2, -1], [-16, -10, 3, -3, 4, -5, -2, -2, 2, -3, 1, -5, -1, -2], [-5, 9, 3, 1, 1, 1, 6, -1, 8, 0, 1, 2, -2, -3], [3, 3, -2, 5, -2, 5, 6, 1, -2, 4, -3, 3, -4, 0], [-10, 7, -5, -1, 1, -6, -2, -1, 0, 0, 1, -4, 3, 1], [1, -5, 3, -2, -1, -3, 2, 1, 1, -3, -1, 0, 1, -2], [-1, -3, 8, -4, 5, -1, -4, -1, 3, 1, -1, 3, -3, -2], [1, -1, -11, 6, -5, -2, 4, 3, -8, 9, 2, -4, 6, 2], [-10, 0, -6, -1, 3, 1, 4, 0, -4, 1, -1, 2, -2, 0], [-8, 2, -8, 4, -2, 2, -4, 5, -11, 5, 0, -3, 3, -1], [-4, -8, 9, -2, 2, 0, -2, -3, 1, -5, -4, -2, -2, 0], [-2, 8, -5, 3, -4, 5, 5, 5, 0, 4, 0, 2, 1, -2], [-10, -1, 10, -13, 7, -6, 0, -7, 14, -8, -3, -3, -3, -5], [2, -5, 10, -6, 6, -5, -7, -6, 4, -8, -3, 4, 0, 0], [5, 1, -1, -3, 1, 4, 8, -1, 3, 2, 0, -2, 3, -3], [2, -7, 8, -5, 1, -6, -2, -1, 4, -4, -1, -3, 2, -2], [-4, -7, -11, 7, -3, 2, 2, 1, -4, 6, 5, -4, -3, -3], [-5, -2, -6, -1, 1, 1, 3, 2, 0, 3, -1, 0, -1, -3], [-16, -5, 6, -5, 0, -3, -5, 0, 1, -7, -3, -2, -1, -1], [-2, -1, -5, 3, -2, 5, -6, 4, -6, -1, 2, 3, 1, 3], [6, -6, -2, 2, -4, 0, 4, -2, 2, -4, 0, 0, -4, -2], [2, -7, -5, 3, -3, 0, 0, -1, -8, -2, -1, 2, -1, 3], [5, 2, -13, 7, -7, 5, 8, 0, -3, 7, -4, -5, -1, 1], [-10, -4, 6, -11, 4, -6, -1, -7, 5, -8, -3, 5, -3, -3], [8, -8, 3, -5, 0, -1, -1, -3, 2, -2, 2, 0, 1, 2], [0, 5, 5, -3, 4, 3, 1, -9, 6, -6, -3, 4, -5, 1], [7, 0, -2, 6, -4, 1, -1, 1, -2, 5, -1, 3, -1, 2], [-1, -1, 1, -7, 0, 1, -6, 3, -2, -6, -1, -2, 7, -1], [-20, -2, 4, 1, 3, -4, -5, 3, -5, -6, -1, 3, 3, 0], [0, -3, 1, -1, 2, 5, -1, -4, 3, -3, 1, 3, 1, -3], [9, 1, 0, 5, -1, 3, 1, 4, -8, 5, 0, -2, 5, 2], [0, -1, -1, 3, 0, -3, -2, 0, -8, -1, 0, -1, 1, 1], [2, -3, -7, 9, -8, 6, 5, 10, -4, 9, 0, -3, 1, 2], [12, -2, 4, 3, 3, 0, 3, 3, 3, 2, -1, 1, 1, 2], [-1, 3, 6, 3, -3, 1, 2, -2, 3, 1, -1, 0, -2, 1], [-10, 3, -1, 2, -1, 0, -6, 0, 0, 0, 5, 3, -1, 1], [-7, 2, -1, 5, -1, -3, -9, 3, -5, 2, 1, 0, 10, 2], [4, -6, 1, 1, 0, -1, -7, 5, 2, -6, 2, 0, 5, 2], [-3, 0, -3, 3, 0, 1, 2, 1, 2, 2, -2, -2, 0, 3], [0, -9, -4, -1, -3, -4, -4, 3, -2, -4, -1, -3, 6, -2], [-7, -8, -5, 6, -4, 1, -3, 7, -8, 5, -1, 0, 2, 3], [-2, -2, 8, -5, 1, 4, 8, -3, 6, -2, -2, -1, -3, 0], [-4, 4, -14, 13, -8, 4, 5, 3, -7, 10, -1, -3, -6, -2], [8, -5, -1, 2, 0, 2, -2, 3, -3, 3, 3, -2, 1, 4], [5, -9, 15, -5, 3, -3, -11, -4, 4, -11, 0, 3, 8, 5], [7, 6, -5, 2, 5, -1, 0, 0, 0, 0, 2, -5, 6, 2], [5, -1, 15, -7, 4, -5, 0, -2, 7, -7, 3, 1, 1, 1], [5, 4, -8, 8, -7, 5, 12, 2, 0, 14, 0, -4, -7, -1], [-6, 0, 4, -5, 2, -2, 9, -5, 7, -4, -1, 1, -5, 3], [-4, -4, 13, -11, 1, 0, 5, -6, 8, -9, -5, 5, -5, 0], [2, -2, -3, 1, -2, 0, 1, 0, 0, 1, -7, -3, 3, -3], [7, 8, -1, 1, -6, -1, -2, 4, -3, 7, 2, -1, -1, 0], [-4, -6, 2, 0, 5, 1, -3, -1, -1, -9, -2, 3, 3, 1], [4, -2, 4, 4, -2, 0, 0, 2, 2, 6, 2, 2, 0, 4], [-8, 2, 9, -3, 7, -7, -10, -2, -2, -5, -1, 1, 1, -1], [0, -1, -4, 0, -4, 3, 4, -4, 5, 4, 4, 0, -2, 0], [-1, -11, -9, 5, -6, 6, 10, -1, -1, 10, 4, 2, -8, -3], [11, -8, -1, -5, 4, -2, 1, -5, 10, -10, -2, 0, 6, -2], [8, -7, 7, 0, 0, 3, 10, -5, 6, -3, -2, 2, -1, -1], [-6, 5, -17, 9, -8, 4, 5, 6, -10, 9, 2, -5, 3, 2], [-4, -10, -4, 3, -5, 2, -7, 1, -5, 2, -1, 1, -3, 4], [-7, 1, 2, -3, -1, 0, -1, -5, -2, 0, 3, 3, -6, 0], [-20, -7, 15, -18, 9, -5, 1, -6, 10, -14, -5, 5, -1, -4], [5, -6, -16, 19, -11, 1, 1, 7, -19, 14, 1, -7, 0, 2], [0, 7, 8, -8, 6, -1, 0, -7, 8, -11, -4, -1, 4, -2], [4, 5, 9, -8, 11, -6, -5, -2, 7, -6, -4, 3, 0, -2], [-5, 5, 4, -5, 4, 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[-3, 5, -5, -2, 4, -3, 1, 9, -4, -1, 4, 2, 8, -2], [15, 0, 9, -1, -1, 8, 3, 1, -2, 8, 0, 10, 0, -1], [-13, 10, -14, 18, -8, 3, -1, 7, -18, 15, -1, 1, 1, 8], [1, 0, -15, 5, -3, -2, -7, -1, -10, 6, 8, -6, -2, 1], [-11, 11, -4, 8, 1, 1, 2, -2, -8, 4, 1, -2, 5, 2], [4, 3, 4, 12, -2, 1, -2, 1, 0, 1, 0, -1, 8, 0], [3, 8, -19, 18, -2, 6, 8, 10, 1, 14, 7, -1, -5, 1], [4, -7, 19, -15, 8, -6, -1, -14, 18, -13, -2, 9, 1, -2], [-14, -5, -21, 15, -14, 13, 9, 15, -14, 20, 5, -8, -8, 0], [-3, 6, -5, -1, -7, -2, 3, 3, -4, 2, -2, -2, -1, -2], [-7, -12, 11, -20, 7, -11, -10, 1, 1, -15, -8, -10, 11, -5], [8, -3, -11, 12, -3, 1, -5, 8, -10, 12, 11, -5, 0, 3], [-6, 15, 9, -8, 0, 7, 10, -3, 14, 3, 2, 6, -4, 0], [-25, 0, -6, -7, 0, -3, -4, -4, 1, 7, 6, -2, -2, -3], [-4, 3, -9, 4, -5, 3, 3, -2, -2, 8, 7, 5, -15, -2], [-11, -10, 21, -11, 7, -5, -11, -5, 3, -12, -11, 6, 1, -1], [15, 9, -3, 5, 0, 12, 4, 3, -5, 16, 4, 8, -9, 2], [-5, -9, 12, -9, 4, -5, -3, -4, 6, -11, -2, -2, -7, -3], [-1, 4, -4, -5, -6, 5, 3, 1, 5, 4, 3, 5, -6, -3], [-2, -22, 12, -5, 5, -8, -4, 0, 5, -11, -8, 2, 7, -4], [7, 2, 15, -6, 8, -8, -16, -6, 7, -12, -11, -1, -3, -1], [-14, -5, 8, -14, -2, 0, 0, 1, 7, -9, -1, 1, 2, 3], [-9, 0, -12, 6, -8, -1, -1, 3, -8, 11, 7, -5, -8, -1], [20, -19, -3, -2, -3, 2, 5, 0, -1, -2, 4, -3, -8, 0], [8, 1, -29, 10, -10, 6, 14, 4, -10, 18, 3, -5, -1, -4], [6, 9, 25, -23, 17, -10, 10, -13, 24, -22, -3, 0, -4, -4], [-8, 9, 11, 0, -1, -8, -11, 1, 2, -3, 10, 6, 0, 8], [19, -10, -9, -4, -5, 3, 20, -4, 4, 12, 4, 1, -2, -4], [-14, 8, -1, -8, 9, -8, -17, 1, 0, -5, 1, -6, 8, -3], [-14, 2, -13, 2, -5, -3, 1, 0, -10, 6, 4, -7, 1, -7], [-19, 6, 2, -12, 12, -3, -5, -3, 2, -7, -5, 1, 11, 2], [18, 19, -17, 13, -4, 11, 3, 7, -2, 10, 3, 0, -6, 0], [-23, 6, 10, -8, 6, -6, -11, 1, 3, -7, -2, 7, 4, -2], [5, -2, 2, 4, -3, 6, 2, -4, -17, 2, -7, 8, 0, 5], [18, -12, -22, 22, -18, 14, 8, 14, -28, 24, 8, 0, 2, 10], [1, -1, -14, 5, -3, -1, -3, 6, -2, 7, 10, -2, 1, 6], [-3, -6, -6, 6, -9, -3, 1, -5, -10, 3, -1, -9, -1, -3], [13, 21, -7, 10, -4, 10, 1, 9, -7, 9, 7, 8, -1, 6], [4, 10, -18, 16, -12, 6, 0, 1, -15, 9, 4, 3, 3, 7], [-1, 23, -7, 17, 3, 5, -1, 0, -6, 13, 4, 1, -9, 6], [11, -16, -9, 10, -7, -5, 2, 12, -12, 2, -4, -15, 2, 4], [16, 7, 10, -11, 8, -10, -7, -8, 16, -13, 0, 4, 10, 3], [-19, 1, 2, -4, -1, -7, 0, -4, 4, 2, -3, -2, -1, -4], [-17, -8, 12, -8, 5, -9, -14, 0, -8, -8, -2, -6, 1, -3], [6, 16, -12, 10, -14, 2, 6, 8, -4, 14, 2, -4, 0, 0], [20, 7, -10, 7, -11, 5, -2, 7, -9, 10, -2, -5, 13, 6], [14, 3, 11, -10, 3, 3, -4, 1, 12, -7, 0, 2, -9, -6], [8, 6, -20, 7, -11, 8, 5, 7, -9, 10, 1, -11, 11, -2], [8, -14, -15, 18, -11, 6, 12, 7, -5, 17, 0, -6, -4, 5], [-16, 2, -8, -4, -8, -2, 8, -4, 0, 2, -10, -6, 8, 0], [1, 8, -5, 6, -4, 8, -1, 6, -16, 12, 0, 1, 8, 0], [-2, -1, -23, 8, -16, 0, 8, 7, -15, 13, 1, -6, 1, 6], [-16, -3, 12, -18, 10, -3, -8, -9, 12, -3, 2, 5, -6, -8], [-15, -5, -12, 9, 1, 3, -8, 2, -13, 3, -5, -10, -2, 5], [-8, 8, 9, -10, 1, -3, -5, -10, 8, -4, 2, 1, -1, -5], [-1, -4, -9, 11, -6, -2, 4, 4, -8, 9, 5, -3, -2, 6], [6, 18, -5, 3, 4, -1, 7, -1, 6, -2, -2, -4, 1, 2], [-8, -13, -14, 1, -9, 7, -2, 7, -1, 6, 4, -5, -4, -5], [12, -3, -21, 19, -15, 12, -2, 13, -22, 24, 3, -8, 3, -1], [-16, 1, 9, -13, -2, -7, 9, -9, 14, -6, -1, -2, -4, -6], [-7, -8, 1, 9, 0, 6, 2, 8, -2, 13, 5, 5, -12, -2], [12, 7, 10, -4, 7, -13, -17, -2, -6, -4, -1, 0, 0, 3], [14, 2, 0, 10, 0, 4, -12, 14, -14, 10, 4, 4, 12, 4], [-6, 8, 12, -2, 0, -4, -16, 0, -10, -8, -8, 2, 8, 4], [-27, 2, 17, -22, 10, -5, 7, -11, 22, -17, -11, 0, 0, -5], [3, 14, -13, 2, 6, 0, 5, 3, -5, -1, 2, -2, -4, 0], [-17, -2, -13, 12, -2, -2, -5, 6, -18, 14, -2, 5, 0, 2], [-3, -10, 2, -3, -1, -7, -4, 4, -9, -1, -2, -8, 10, -4], [6, -11, 3, -7, 4, -3, 11, -6, 11, -5, 1, -1, -11, -5], [-6, -4, 1, -5, 7, -1, 6, -10, 6, -7, -1, -3, -5, -3], [13, -9, -2, -1, -3, 1, 11, 2, 12, -3, 2, -10, 7, -8], [1, 11, -2, 0, -2, 5, -1, 0, 9, 8, 8, -6, -3, -1], [9, 12, -3, -2, -3, 6, -10, 4, -3, 6, 11, 1, -7, 2], [21, 0, 27, -17, 6, -3, -6, -7, 6, -10, -2, 8, 8, 7], [-8, -2, 20, -8, 6, -4, -24, 2, 6, -16, 2, 0, 4, 2], [6, 11, -31, 11, -11, 3, 0, -1, -9, 12, 2, -8, -4, 1], [-4, -27, -30, 19, -16, 5, 1, 14, -19, 23, 13, -4, -5, 1], [9, -12, 11, -12, 4, -11, -9, 5, 14, -9, 5, -8, 8, -5], [6, -7, 11, -13, 3, 5, 10, -5, 11, -8, -4, 2, 0, -1], [10, -11, 9, -7, 4, -7, 9, -8, 19, -7, -9, -15, 5, -5], [22, -17, -28, 23, -15, 8, -4, 15, -26, 22, 7, -5, 4, 4], [21, 2, 21, -15, 16, -9, -16, 1, 4, -16, -4, 14, 6, 13], [-30, -14, -5, 8, -7, 1, 5, 2, -2, 10, 0, 3, -9, -7], [-19, -10, 21, -11, 8, -14, -8, -4, 8, -19, -11, -3, 12, 4], [-11, 7, -29, 18, -11, 1, -2, 15, -19, 19, 1, -10, 3, 0], [13, -15, -3, -4, -7, 5, 1, 3, -2, 9, 8, 6, -6, 3], [16, 6, -3, -3, -8, 10, 15, -1, -1, 2, -5, 2, -10, -6], [-2, 6, 4, -5, 9, -10, -6, 2, 7, -5, -4, 4, 1, 0], [6, -4, -2, 13, -9, 2, 3, 11, -1, 4, 7, -1, -7, 6], [1, 0, 5, -2, -2, 4, 12, -12, 15, 2, 1, -1, -11, -1], [-8, -4, -2, 10, -4, 2, 6, -3, 1, 9, 4, 1, -3, 7], [-6, -9, 1, -2, 3, -11, 0, -13, 2, -13, -6, -14, 9, 2], [10, 9, 5, 1, 0, -13, -7, -5, -4, -6, -5, 0, 11, -1], [6, -3, -10, 12, -11, 1, 4, 13, -12, 11, 4, -3, -2, -1], [10, 9, 17, -9, 9, -3, -18, -7, -5, -14, -10, 6, 4, 1], [-11, 7, -11, 4, -3, 6, -15, 2, -10, -6, 1, 5, -4, 2], [12, -8, 29, -15, 10, -7, -15, -17, 6, -14, -2, 16, 1, 2], [-7, 8, -8, 8, -3, 0, 4, 6, -7, 8, 11, 4, 2, -1], [-5, 3, 20, -12, 2, -5, 0, -9, 11, -7, -1, -7, -4, -4], [-27, 6, 25, -19, 25, -6, 1, -19, 30, -24, -4, 10, -7, -8], [-17, 3, -2, -7, 3, -11, -19, 10, -8, -7, -4, -16, 17, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9405_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9405_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_9405_2_a_bq(:prec:=14)
    chi := MakeCharacter_9405_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_9405_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_9405_2_a_bq( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9405_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 25, 142, -142, -567, 240, 766, -193, -467, 80, 137, -15, -19, 1, 1]>,<7,R![4, -88, -160, 4244, 9153, -8428, -17617, 2602, 8260, 1228, -974, -272, 18, 12, 1]>,<13,R![43200, 19008, -337008, -120176, 345564, 151144, -119660, -63816, 13907, 10416, -11, -582, -40, 10, 1]>,<17,R![-64, 128, 22928, -94384, -16236, 322336, -142268, -103352, 41837, 13598, -2935, -942, 6, 16, 1]>],Snew);
    return Vf;
end function;