
// Make newform 240.2.y.e in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_240_y();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_240_y_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_240_2_y_e();".
// 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_240_2_y_e();".
// 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 := [256, -768, 896, -320, -416, 624, -264, -148, 233, -74, -66, 78, -26, -10, 14, -6, 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, 0], [1792, -5056, 4480, -240, -3360, 3188, -424, -1415, 1110, -2, -530, 342, -26, -94, 62, -15], [64, -1760, 2624, -1232, -908, 1834, -892, -409, 734, -250, -202, 238, -70, -36, 42, -13], [-256, 3584, -5408, 2544, 2064, -3856, 1822, 911, -1538, 502, 438, -498, 142, 78, -88, 27], [-1664, 3680, -3056, -80, 2520, -2162, 207, 1001, -752, -24, 368, -232, 16, 66, -43, 11], [-1120, 2272, -1640, -308, 1594, -1158, -53, 646, -382, -78, 226, -114, -8, 40, -21, 4], [5696, -11360, 7744, 1904, -7876, 5462, 492, -3173, 1758, 454, -1098, 526, 50, -192, 98, -17], [-12032, 29184, -25312, 432, 19808, -17864, 2070, 8165, -6254, -134, 3034, -1918, 114, 538, -348, 81], [-12544, 31296, -27584, 912, 21248, -19540, 2428, 8807, -6830, -86, 3290, -2094, 130, 582, -378, 87], [7104, -17216, 15088, -448, -11660, 10772, -1407, -4820, 3786, 10, -1806, 1170, -82, -322, 213, -50], [18048, -39616, 30688, 3120, -27464, 21852, -466, -11253, 7358, 950, -4042, 2222, 38, -718, 408, -81], [-18560, 43328, -35872, -976, 29624, -25404, 2006, 12197, -8758, -558, 4466, -2662, 82, 790, -484, 105], [-29952, 63232, -47008, -7024, 43712, -33152, -442, 17785, -11014, -1854, 6306, -3318, -134, 1114, -612, 117], [18432, -42752, 35072, 1408, -29248, 24788, -1756, -11991, 8494, 598, -4378, 2590, -66, -774, 474, -103], [40448, -93888, 76992, 3376, -64304, 54212, -3584, -26407, 18534, 1422, -9602, 5638, -122, -1694, 1030, -223], [-25152, 56544, -45120, -3120, 38796, -31850, 1520, 15875, -10826, -1058, 5742, -3290, 26, 1016, -602, 127]];
    Rf_basisdens := [1, 256, 128, 256, 128, 32, 128, 256, 256, 128, 256, 256, 256, 128, 256, 128];
    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_240_y();"
function MakeCharacter_240_y()
    N := 240;
    order := 4;
    char_gens := [31, 181, 161, 97];
    v := [2, 3, 4, 3];
    // 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_240_y_Hecke();"
function MakeCharacter_240_y_Hecke(Kf)
    N := 240;
    order := 4;
    char_gens := [31, 181, 161, 97];
    char_values := [[-1, 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], [1, 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]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, 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+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, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 1, 1], [-1, -1, 0, 1, 1, 0, 1, 1, 0, 1, -1, 0, 1, 0, 0, -1], [0, 1, 0, -1, 0, 0, 0, 0, -1, -1, 0, 1, 0, -2, -1, -1], [0, 0, 1, 1, 0, 1, 0, -2, 2, 0, 2, 1, 1, 0, 0, 1], [-1, 1, 0, 1, 0, 0, 0, 2, 0, 1, -2, -1, 0, 0, -1, -1], [3, 1, 0, 1, 1, 1, 1, -2, 2, -1, -1, 0, 0, 4, -1, 1], [0, 0, 0, -1, -1, 1, 0, -1, 0, 0, -1, 1, -1, 0, 1, 0], [2, 1, -2, 1, 1, 2, 1, -5, 1, -1, -1, 2, -3, 0, -2, 1], [0, 0, 0, 3, 3, -3, 1, 2, 4, 3, -5, -4, 2, 3, -3, -3], [0, 0, -1, 1, 2, 1, 0, -4, 0, 0, 0, 1, 1, 0, -2, -1], [-1, -1, 2, -2, -4, 0, -3, 1, 0, -1, 6, 1, -1, -2, 4, 3], [0, 0, -2, -1, 1, -3, 2, 3, -2, 2, -5, -1, 3, 2, -1, -4], [-1, -1, 0, 0, 0, -2, 1, 1, 2, 1, 2, -1, 1, 2, 0, 1], [3, -1, 1, 1, 0, -1, 2, 0, 0, 0, 2, -1, 1, 1, 0, -1], [-1, 2, 2, 5, 0, -2, 1, 3, -1, 0, -2, 0, -1, 0, -1, 0], [-1, -2, 0, 3, -1, 1, 0, -1, -3, 2, 3, -1, 1, -2, 1, 4], [2, 2, -2, -2, 0, 0, 0, 0, 2, 0, -2, 0, 0, 2, 0, -2], [-1, -3, 2, -3, 1, -3, -3, 6, 0, 1, -1, 0, 0, 2, 1, -3], [5, 6, 0, -3, 1, 1, 0, -3, 5, 0, 3, 1, 1, 0, 1, 0], [-4, -2, 0, 1, -1, 1, -1, 0, 0, -1, -1, 0, 0, 3, -1, 1], [1, -1, -2, 2, 0, 2, 3, -5, 0, -1, 2, -1, 1, 2, 0, 3], [-3, -1, 0, -3, -1, 1, -3, 4, 0, 3, -1, 2, -2, -2, -1, -3], [3, 0, 6, 5, -1, -5, 0, 7, 3, 0, 1, -1, 1, 0, 1, 0], [-3, 0, -6, -1, -1, 2, 0, -4, -5, -2, -3, 1, -2, 0, 0, 2], [-5, -3, 0, -2, -4, -1, -1, 2, 0, 3, 4, 1, 0, -2, 3, 1], [-1, 1, 0, 3, -3, 3, 1, -2, 0, -1, 1, 0, 0, 0, -3, 1], [3, 4, 0, 4, 0, 2, 2, -4, 1, -2, 0, 2, -4, 0, -2, 2], [-2, 0, 0, -7, -2, 0, -1, 1, -2, 0, 0, 2, -5, -4, 5, 2], [-3, 1, 0, -4, -4, 5, -1, -4, -2, -1, 2, 5, -4, -6, 3, 5], [1, 4, -6, 1, 2, 4, 3, -9, -1, -2, -2, 4, -3, 0, -1, 2], [0, 0, 0, 2, 3, -3, 5, -2, 6, 0, -7, -3, 2, 6, -2, -4], [-3, -3, 0, -1, -5, -1, -3, 2, -6, -3, 1, -2, -2, 0, 1, 3], [2, 5, 0, -5, -1, 2, 1, -3, 1, -5, 3, 6, -1, -2, -4, 1], [-4, 2, 4, 0, 2, -6, -3, 13, 0, 1, 0, -5, 5, -3, 2, -5], [6, 4, 1, -10, 1, -2, -2, -1, 0, -2, 5, 2, -2, -2, 1, 1], [4, 0, 0, 5, 1, -1, 2, 1, 0, 2, -3, -1, 1, -2, 1, -2], [3, -3, 2, 2, -2, 0, 1, -1, 0, -3, 4, 1, -1, 0, 2, 3], [0, 0, -7, -6, 1, 0, 1, -2, -4, 1, -7, 1, 3, 1, -1, -8], [2, -3, 0, 6, 1, -1, 2, -1, 1, 3, 1, -4, 3, 6, 0, 3], [-3, -14, 0, -2, -2, 0, 0, 4, -3, 0, 2, -2, 4, 0, 4, 0], [3, 3, 4, 3, -1, 5, -5, -6, 6, -1, 7, -2, -4, 2, 1, 5], [5, 4, 0, -4, 0, -2, 0, 2, 1, -4, 2, 4, 2, 2, -6, -6], [8, 8, -1, 5, 6, 3, 1, -7, 10, -1, -2, 0, -2, 7, -6, 0], [2, 2, 0, 6, 6, 0, 5, 1, -4, 1, -8, -3, -1, 3, -6, -1], [-3, 1, 8, 1, -3, -5, -3, 10, -2, 1, 3, -4, 2, 0, 1, -1], [7, -5, 0, -3, 1, 0, 1, -11, 2, -5, 9, 6, -1, 0, 4, 5], [0, 0, 2, 3, 1, 5, 0, -7, 0, 2, 1, -1, 3, 2, -1, 0], [1, 2, 0, -1, 3, 1, 2, -3, 3, -2, -3, -1, -3, 2, 1, 4], [1, -1, -2, 4, 5, -3, 2, -3, 2, 1, -5, 4, -1, 0, -2, -1], [9, -1, 4, 3, 1, -5, 1, 8, 0, -1, 3, -6, 6, 2, 1, -3], [8, 8, -6, -10, 0, 6, 0, -18, 0, -4, 2, 8, -8, -4, 0, 10], [4, 3, 0, 3, 0, 4, 8, -12, 1, -3, -6, 3, 2, 2, -5, 1], [-3, -3, -2, 4, 3, 3, 6, 3, 0, 3, -3, 0, 5, 0, 2, -3], [7, 1, 0, 0, 4, -4, -3, 7, -4, -1, -4, -3, 1, 2, -2, -9], [-4, -1, -8, 0, 6, 3, 5, -4, -1, 3, -6, 3, 4, 0, 1, -3], [-4, -4, -4, -2, 2, 4, 3, -7, -8, 1, -4, 3, 5, -3, -2, -5], [-2, -2, 3, -3, -6, -7, -6, 8, -8, 2, 6, -3, 1, 0, 6, 1], [-3, -3, 4, 7, 3, -3, 5, 4, -2, 3, -3, -2, 4, 0, -3, 1], [-4, -4, 6, -2, -8, -4, -4, 12, 2, 4, 2, -8, 0, 0, 8, 2], [6, 6, -5, -2, -3, 8, -3, -4, 0, -1, 1, 1, -1, -5, -3, 6], [-8, -8, 0, -4, -4, -4, -2, 4, -4, 4, 0, -2, 6, -4, 4, -4], [-1, 5, 0, -5, -3, 3, -5, 4, 0, 1, 1, 2, -2, -6, -3, -1], [3, 0, -2, 0, 2, 4, 0, -12, 1, -2, 0, 4, -8, 0, -6, 2], [10, -4, 1, -11, 2, -3, -3, 1, 0, -1, 4, 2, -2, 5, 2, 0], [5, -3, 0, -1, 3, -7, -1, 8, 2, 3, -1, -6, 0, 10, 3, -3], [-5, -4, -8, 1, -3, 7, 4, -1, -7, -2, -3, -1, 1, 0, 3, 2], [-10, -6, -4, 7, 5, -1, -2, 9, 0, 6, -9, -5, 5, 0, 5, -2], [-3, -6, 2, 5, -5, 1, 0, 9, -3, 0, -3, -5, -1, 0, -1, 0], [2, -2, 0, -1, -4, 2, 1, -3, 4, 2, 6, 2, -1, 8, 3, 8], [-4, -4, 2, -4, -6, -4, 2, 6, -4, 4, 2, 2, 10, 0, 6, -2], [-5, 1, 0, -12, -2, -5, -7, 12, -2, -1, 4, 3, -4, -8, 3, -7], [-2, 4, 9, -4, -5, -4, -4, 13, 0, 0, 5, 0, 0, -4, -5, -9], [-3, -9, 8, -3, -3, 3, -1, 4, -2, 1, 9, -4, 2, 0, 1, -1], [-14, -2, 0, 3, -1, -1, -2, 13, -8, 6, -11, -7, 1, 0, -5, -6], [-5, 0, 0, -6, 6, -9, -4, 13, -5, 0, -2, -6, -3, -10, 3, -6], [-2, 2, 3, -10, 3, -6, -8, 13, 0, 0, 1, -2, 2, -2, 3, -3], [2, 2, -8, -7, 5, 3, 2, -11, 0, 2, 1, 3, -3, -4, 5, 6], [5, 5, 4, -7, -7, 3, -3, -2, 0, -5, 7, 4, -4, 0, -7, 1], [-4, 1, 0, -3, -6, 6, 4, -10, 3, -1, 6, 7, -2, -2, 1, 11], [-7, 0, -6, -5, 1, 3, 0, 7, -1, 6, -7, -5, 7, 0, 1, -6], [2, 2, -8, -4, 4, -4, 0, 0, 0, 0, -8, 0, 0, 2, -4, -8], [3, 4, 0, 9, 11, 1, 6, -7, 9, -4, -7, -7, 1, 8, -5, 6], [4, 4, 0, 0, 0, -6, -3, 5, 4, -1, 2, -1, -3, 3, 0, 1], [2, 2, -4, 5, 9, -9, 2, 7, 4, 6, -17, -11, 1, 8, -9, -10], [0, 0, -6, -9, -3, -1, -6, -5, -8, -2, 5, 5, 1, -2, 3, -4], [-5, -4, 0, 9, 5, -3, 6, -3, 5, 4, -11, -9, 3, 4, 1, -2], [6, 5, 0, 3, 1, 4, 3, -7, 13, -5, -3, 4, 3, 14, -8, -3], [-7, 1, 0, -5, -9, 4, -5, -9, -16, -9, 9, 0, -9, 0, 0, 9], [-9, -9, 0, 2, 2, -10, -3, 9, 2, 9, -10, -13, 5, 0, -2, -9], [3, -3, -4, 7, -3, 7, 1, -6, 0, -1, -3, 0, 0, 4, -3, 5], [13, 5, 4, -3, -3, 4, -1, -5, 10, -3, 9, 0, -3, 0, 0, 3], [-8, -5, 0, 2, -1, -3, 4, -1, 5, 5, -3, -4, 1, 2, 4, 1], [-7, -3, 0, -7, -3, -5, -13, 18, -2, 3, 5, 0, -2, -6, 5, -11], [3, 7, 0, 0, 6, 4, 1, -5, 10, -7, -2, 1, -3, 6, -4, 5], [15, 8, 0, -2, 2, 7, 0, -11, 15, 0, 2, 2, -7, 0, -7, 0], [4, 4, -4, 3, 7, -5, -4, -9, 4, 2, 1, 1, 5, 6, -7, -6], [0, 0, 0, -1, -1, 7, 6, -5, 0, 0, -1, 5, 5, 0, 1, 0], [-1, -4, 0, 12, 0, 6, 0, -6, -5, 4, -4, -4, 2, -2, 2, 4], [-14, -14, 6, 9, 3, 3, 2, -3, -6, 2, 1, -3, 1, -12, -3, 4], [10, 2, 3, -6, 1, -4, -1, 2, 0, -1, 3, 1, -1, -1, 1, -2], [0, 8, 6, -5, -1, -5, -4, -3, 0, -10, -1, 3, -3, 2, -1, 4], [-8, -4, 4, -3, 5, -9, 0, 15, 0, 8, 1, -3, 3, -4, 5, -12], [-1, 1, 0, -13, -11, 3, -5, 2, -8, -1, 15, 12, -4, -10, 3, 5], [-13, -6, 8, 5, -1, -11, 0, 29, -5, 8, -5, -9, 15, 0, 7, -8], [-9, -7, 2, 11, -5, 3, 1, 8, 0, 1, 1, -6, 6, 6, -5, -3], [-12, -4, 0, -3, -2, -2, -5, 7, -8, 4, 8, -2, -3, -16, 7, 6], [-2, -2, 4, 0, -4, 0, 0, 8, 0, 4, -4, -8, 0, 2, 4, 0], [-1, -1, 8, 1, -7, 5, -1, 2, 10, -5, 13, 4, -6, -6, 7, 13], [11, 3, 0, -2, 10, -11, 7, 4, -8, -3, -2, -7, 2, 0, -5, -1], [20, -4, -1, 4, 5, -4, 8, -15, 0, -8, 1, -2, 2, 12, 5, 9], [-4, -2, 0, -3, 2, -6, -7, 13, -2, 2, 0, -4, -1, -4, 3, -8], [7, 17, -12, -7, 11, 3, 1, -14, 12, 5, -9, 6, 0, 0, -5, -5], [6, -4, 4, 11, 1, -5, 1, 12, 0, 5, -7, -4, 4, -1, 1, -9], [-19, -3, 6, 0, -4, -2, -7, 11, 0, -5, 0, -3, 3, 8, -4, -1], [-14, -6, -4, 17, 1, 3, 2, 9, 0, 6, -7, -9, 9, 0, 1, -2], [-5, 7, -2, 8, 2, -14, -3, 13, -2, 3, -16, -1, 1, 0, -2, -3], [1, 1, 3, -3, -6, -7, -8, 6, -16, 0, 10, -1, -1, 1, 6, 3], [8, -1, 0, -8, -3, -1, 0, 1, -15, 1, 3, 2, -5, -6, 6, 5], [-7, 0, 2, 7, 5, -3, 0, 21, 7, 14, -19, -9, 5, 0, -9, -14], [-3, 0, 0, -7, -5, 3, -8, 5, -9, 0, -1, 5, -5, -12, 5, -6], [-1, -1, -5, -1, 4, 3, 2, -6, 12, 0, -6, 1, 1, -1, -4, -5], [7, 7, -8, 8, 16, 4, 13, -5, 6, 1, -24, -3, -1, 8, -16, -9], [-11, 3, -8, 1, 1, 3, 1, -10, -14, -3, -5, 4, -6, 0, -3, 3], [-11, -4, 0, 3, 1, -4, -2, 6, -9, 4, 3, -5, 2, -16, 2, 0], [3, 4, 0, 5, 1, 5, 6, -11, -5, -4, -7, 3, 3, -6, -7, -2], [-18, -6, -2, 10, 4, -2, 4, 8, 0, 10, -6, -4, 4, -4, 4, -8], [-3, 5, 0, -5, -1, -4, -5, 3, -2, 1, -1, -2, 1, 0, 0, -1], [-8, -8, 1, 1, 0, -3, 4, 8, 8, 4, -8, -5, 3, -4, 0, -3], [-5, 3, 0, 7, 3, 15, -3, -12, -6, -3, -1, 0, -8, -14, 5, 19], [-10, 2, 0, 11, 7, -1, -2, 3, 20, -2, -1, -5, 7, 8, -9, -6], [-6, -6, 16, 9, -7, -5, 1, 12, -4, 3, 13, -2, 4, -3, 7, 13], [6, 6, 1, 5, 4, 13, 10, -10, 0, -2, -4, 5, 1, 4, -4, 3], [2, 2, 4, 3, -1, 7, -2, -7, 20, 0, 5, -1, -1, 2, 1, 4], [11, 11, 2, -7, -9, -7, -5, 8, -8, -1, 11, 4, 2, 10, 9, 3], [14, -4, -7, -1, 4, 3, 5, -17, 0, -3, 0, 2, -2, 7, 4, 10], [10, 10, -4, -9, -5, 7, -2, -1, 8, -4, -1, 1, -7, 6, 5, 0], [-32, 6, -11, -3, -2, 13, -7, 1, 0, 13, -10, 8, -8, -19, -2, -2], [7, 7, 0, -2, -4, 4, 1, -9, 0, -7, 6, 1, -1, 0, -4, 7], [5, 7, 0, 7, 3, 13, -3, -10, -2, -7, -1, 4, -2, -4, -5, 7], [-20, -11, -6, 3, 1, -8, -5, 15, -13, 7, -21, -6, 1, 0, -6, -7], [-1, -1, 0, -8, -8, -1, -7, 8, -6, 1, 2, 7, 0, -6, 1, -13], [16, -8, 14, 4, -2, -12, 0, 18, 0, -2, 4, -6, 6, 10, -2, -12], [5, 2, 8, 12, -4, 4, 6, 10, 5, 0, 2, -4, 0, 0, 0, 0], [-8, -8, 0, -2, -2, -6, -6, 2, -8, 2, 4, -2, 2, -6, 2, -2], [4, -4, 7, 14, -5, -2, 6, -9, 0, -10, -3, 0, 0, 14, -5, 3], [8, -2, 0, -20, -1, -15, -3, 18, -8, 2, -3, -1, -6, 2, 8, -14], [9, 1, 4, -3, 3, -3, -3, -8, 8, -1, 5, 4, -4, 0, -3, 1], [-7, -1, 6, 4, 8, -2, 3, 11, 4, 11, -6, -3, 7, 0, -4, -11], [-2, -24, -5, 8, 5, 0, 12, -13, 0, 0, 3, -4, 4, 24, 5, 5], [21, 15, 10, -5, -5, -5, -9, 14, 0, -9, 5, -4, 4, -6, -5, -1], [4, -12, 4, 7, 3, -7, 10, 7, 0, 2, 11, -11, 11, 10, 3, -6], [-5, -13, 6, -4, -4, -2, -3, -11, -14, -9, 16, 5, -5, 0, 4, 9], [-18, 0, -6, 15, -5, 11, 4, -11, 0, -2, -5, -1, 1, 2, -5, 8], [2, 2, 4, 4, 0, 4, -6, -6, -8, -6, 12, 2, -10, -4, 0, 10], [6, 6, -8, -7, 1, 3, -1, -16, 20, -5, 9, 16, 6, 1, -1, -3], [-4, -4, -4, -3, 1, -3, 4, 9, -16, 2, -13, -5, -1, -2, -1, -6], [-5, -4, 0, 12, 0, -2, 2, 0, 17, 4, -6, -4, 10, 16, -6, -12], [5, 2, 0, 10, 12, -11, 16, -5, 19, -2, -8, -10, 11, 22, -13, -4], [6, 6, 8, -4, -12, 12, -10, -12, 0, -10, 30, 12, -8, -4, 12, 18], [7, 13, 4, -5, -14, 0, -8, 6, 0, -7, 8, -7, -4, 0, 3, 7], [-1, -15, -4, -7, 11, 4, 3, -21, 0, 1, 5, 10, -1, 0, -2, -1], [14, 1, 0, -23, -12, 0, -2, 2, -23, -1, 8, 13, -10, -10, 9, 3], [11, 9, 0, -11, -10, -2, -10, -18, -4, -15, 16, 5, -12, 0, 3, 15], [-18, -2, 0, 1, -1, -1, -2, 3, -16, 2, -3, -1, 1, -32, 1, -6], [2, 2, 4, 0, -4, 2, -3, 3, 24, -3, 6, -1, -7, -1, 4, 7], [6, 6, -19, -7, 12, 9, 4, -18, 2, -12, -10, 13, -11, -6, -12, -7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_240_y_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_240_2_y_e();".
// 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_240_2_y_e(:prec:=16)
    chi := MakeCharacter_240_y();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_240_2_y_e();".
// 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_240_2_y_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_240_y();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2304, -23040, 115200, -251904, 314752, -202368, 56832, 7360, 7856, -7648, 2432, 1264, 392, -32, 8, 4, 1]>,<11,R![952576, 31232, 512, -36096, 1229440, -4608, -128, 3264, 335152, 4864, 32, -400, 1288, -8, 0, 0, 1]>],Snew);
    return Vf;
end function;