
// Make newform 240.2.s.c in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_240_s();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_240_s_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_s_c();".
// 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_s_c();".
// 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 := [1024, 0, -512, -512, 448, 512, 96, -288, -168, 80, 136, 40, -42, -36, 6, 16, 7, -4, -2, 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, 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, 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, 512, 512, -448, -512, -96, 288, 168, -80, -136, -40, 42, 36, -6, -16, -7, 4, 2, 0, -1], [5632, -2816, -8448, -7040, 2976, 5536, 1024, -3808, -3332, -152, 1426, 768, -364, -616, -199, 94, 104, -8, -33, -18], [-17920, -5120, 11776, 17728, -736, -11680, -7264, 3800, 6636, 2008, -2086, -2082, -64, 946, 497, -67, -204, -30, 39, 31], [18432, 5632, -12032, -17664, 1024, 12192, 7264, -4160, -6832, -1892, 2296, 2146, 12, -1012, -508, 89, 220, 32, -44, -33], [9472, 640, -8320, -10656, 1472, 7152, 3712, -3220, -4336, -1030, 1480, 1254, -96, -673, -318, 59, 136, 13, -30, -23], [-5632, -11264, -8448, 1024, 5440, 992, -4448, -4816, -1184, 1356, 1096, -266, -836, -464, 12, 131, 28, -44, -36, -11], [28672, 24064, -128, -21760, -7936, 11424, 15248, 3232, -5696, -4436, 652, 2786, 1376, -364, -594, -123, 192, 104, 10, -21], [31744, 17920, -10752, -27648, -3840, 16608, 14656, -1664, -9008, -4156, 2208, 3326, 780, -1052, -760, -1, 284, 80, -32, -39], [-10752, -14336, -6656, 5440, 6176, -1408, -6720, -4600, 148, 2008, 790, -746, -952, -310, 119, 129, -12, -54, -31, -5], [32256, 23296, -5248, -26176, -6336, 14848, 15952, 760, -7800, -4504, 1624, 3282, 1120, -790, -720, -47, 264, 98, -16, -33], [-16896, -9216, 5632, 14464, 1440, -8960, -7680, 1120, 4852, 2096, -1298, -1808, -380, 588, 407, -10, -160, -44, 17, 22], [-19968, -15872, 1024, 15360, 5088, -8064, -10176, -1712, 4108, 2944, -574, -1884, -844, 312, 409, 68, -136, -64, -1, 16], [39936, 33280, 1536, -29056, -11200, 14368, 20384, 4688, -7320, -5860, 764, 3622, 1804, -480, -770, -161, 260, 132, 10, -31], [-10240, 5632, 17536, 14208, -5440, -11168, -2480, 7392, 6816, 588, -2764, -1598, 632, 1204, 426, -163, -200, 8, 62, 35], [-5120, 10752, 16896, 9216, -7680, -8736, 928, 8160, 5440, -548, -2688, -990, 956, 1132, 280, -199, -164, 32, 64, 31], [-24064, -18944, 2816, 19200, 5504, -10944, -12464, -1360, 5544, 3556, -1000, -2450, -976, 484, 532, 59, -184, -80, 4, 21], [110080, 39424, -65152, -105024, 2144, 70208, 45936, -20280, -39068, -12364, 12266, 12724, 816, -5454, -2983, 376, 1228, 214, -217, -180]];
    Rf_basisdens := [1, 1, 1, 256, 512, 512, 512, 256, 256, 512, 512, 256, 512, 256, 256, 512, 256, 256, 256, 512];
    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_s();"
function MakeCharacter_240_s()
    N := 240;
    order := 4;
    char_gens := [31, 181, 161, 97];
    v := [4, 3, 4, 4];
    // 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_s_Hecke();"
function MakeCharacter_240_s_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, 0, 0, 0, 0, 0, -1, 0, 0, 0, 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], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 1, 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, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 1, -1, -1, 0, 0, -1, -1, 1, 1, 0, 2, -1, -1, 0, -1, -1, 0, -1], [0, 1, 1, 0, 0, 2, 0, 0, 1, 0, 1, 0, 0, 0, -1, -1, 0, 0, 1, 0], [0, -1, 1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0], [-1, -2, 0, 0, 0, 1, 0, 0, 1, 0, 0, -1, -1, 1, 0, 1, 0, -1, 0, 1], [0, -1, -1, 1, 1, 0, 1, 0, 1, 1, -1, 0, -1, 0, -1, -1, 0, 0, 0, -1], [1, 1, -3, 1, 2, 1, 2, 2, 1, 0, -1, -1, -1, 1, -1, -2, 2, 1, 1, 1], [1, 1, -1, 1, 1, 0, -1, 0, -1, 2, 1, -1, 1, 0, -1, -1, 0, -1, -1, -1], [-1, 0, 0, -2, -1, 0, 0, 0, -2, 0, 0, 3, 3, -1, 0, -1, -1, 1, 0, -1], [1, -1, -2, 1, 1, 0, 1, 1, -1, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 2, 2, -2, 0, 1, 2, -2, 0, 0, 0, 0, 0, -1, 0, 0, 0], [-1, 2, 1, 1, -1, 0, 1, -1, 0, 1, 1, 2, 0, -2, -1, -1, 0, -1, 0, -1], [-1, 0, 1, -1, 0, 1, 0, 1, 1, -1, 1, 0, 0, 1, -1, 0, 1, 3, 2, 1], [-1, -2, -1, 1, 1, 0, -1, 1, 0, 1, -1, 2, 0, 2, 1, 1, 0, 1, 0, 1], [0, 1, -1, 2, 2, -2, 2, 2, 1, 0, -1, 0, -4, 4, 1, 1, 2, 2, 1, 2], [1, 1, 0, 0, 2, 0, 0, 1, 0, -2, 0, -1, -1, 0, 2, 0, 1, 2, 1, 2], [-2, -2, 2, 0, -4, 0, 0, -2, 1, 0, 2, 0, 0, 0, -2, 2, 1, 0, 0, 0], [-1, 1, 1, -3, -1, 0, -1, 0, -2, 0, 1, -3, 5, 0, -1, -1, -3, -1, 1, -1], [1, 2, 1, -1, 1, 0, 3, 1, 0, 1, 1, 0, -2, 2, 3, 1, 2, 1, 0, 1], [1, -1, 2, 0, 1, -2, 0, -3, 0, 3, -2, 2, 2, 3, -2, 1, -2, -1, -1, -1], [-1, 0, -1, 1, 3, 0, -3, 1, -1, -3, -1, 0, -2, 0, 1, -1, 1, 1, 0, 1], [0, 4, -2, 0, 0, 0, -2, 2, 1, -4, 0, 2, -2, 0, -2, -2, 1, 0, 0, 0], [3, -2, 1, -1, -1, 2, 1, 1, 2, -1, 1, -2, -2, 0, -1, -1, 2, 3, 2, 1], [2, 0, 0, 0, 0, 2, 0, 0, 0, -2, -4, 0, 0, 0, 0, 0, 2, 0, 0, 0], [-1, 2, -1, -1, -3, -2, -2, -1, -1, 1, -1, 0, 2, -3, -3, 0, -1, -1, -2, -1], [1, 2, 1, -1, -1, 2, 1, 1, -1, -3, 1, 0, 0, -2, -1, -1, 1, 1, 0, -1], [1, 3, -4, 2, 0, 0, 2, 3, 0, -2, 0, -1, -1, 0, 2, -4, 1, 0, -1, 0], [-3, -2, 2, 0, 2, -3, 0, -2, 1, 2, -2, -1, -1, 3, -2, 1, 0, 1, 0, -1], [0, -2, -2, 0, -2, 2, 1, 0, -2, 0, -2, 0, 0, -1, 2, 1, -2, -4, -2, 0], [-1, -4, 0, 3, -3, 0, -1, 0, 1, 0, 0, -1, -3, 0, 0, 0, 1, -3, -2, -1], [-1, 1, 2, -2, -4, -3, -6, -3, -2, 1, 0, 0, 2, -3, 0, 3, -4, -1, -3, -1], [-2, -1, 3, -1, -1, 2, -1, -4, 2, 5, -3, -2, 1, 2, 1, 1, -1, -2, 0, -1], [-5, 3, 1, 1, -1, -2, 3, 0, -1, 4, -1, 3, 1, -4, -3, -3, 2, -1, 1, -1], [1, 3, -2, 0, 1, 0, -2, 1, -3, -1, 4, 0, -2, -1, 0, 1, 1, 1, -3, 1], [2, 1, 1, -2, 0, 0, 4, 0, 1, 2, 3, -3, 2, 0, 1, 1, 0, 3, 4, 0], [3, 2, -3, 1, 1, 0, -3, 1, -3, 5, 1, 0, 2, 0, -1, -3, -1, -3, -2, -3], [0, 2, 0, 0, -3, -3, 1, 0, 0, -6, 0, 2, -2, -1, 0, 1, 2, 0, -2, 0], [-2, 3, -2, -2, 4, 0, 4, 1, 1, -3, 0, 1, 1, 0, 0, -2, 0, 4, 3, 0], [5, -2, 2, 1, 3, 0, -3, 0, -1, 4, -2, -3, -1, 0, 2, 2, -1, -1, 0, 1], [-1, 3, 0, 2, 2, -4, 2, 3, 2, 0, -4, -1, -7, 6, 2, 2, 5, 4, 1, 2], [5, -1, -1, 1, -1, 2, 1, 0, 0, 0, -1, -1, -1, 0, 1, 1, -1, -3, -1, 1], [4, -2, 0, -2, 0, 2, -4, 2, -2, -2, -4, 0, 0, 2, 0, 2, 2, 2, 2, 2], [3, -3, 1, -1, -1, 2, -1, -2, -1, -2, -5, 1, 3, 0, 1, 1, 0, -1, -1, -1], [1, -6, -2, 4, -1, -2, 0, 0, 1, 6, 0, -1, -1, 1, 0, -1, 0, 1, 2, 1], [-4, 0, 1, 1, -1, 0, 3, -1, 2, -4, 3, -1, 0, 0, -1, 1, 2, 0, -1, 1], [-2, -2, 0, 0, -6, 4, -5, -2, 2, 2, 4, 0, 0, -1, 0, -1, 2, -2, -2, -2], [-1, 4, -1, -3, -1, 0, 1, 1, -1, -3, -1, 0, 2, 0, 1, -1, -1, 5, 4, 1], [3, -3, -2, -2, 2, 8, -2, 3, 0, -4, 2, -1, 5, -2, -4, -4, -1, 0, 3, 2], [1, -3, 0, 2, 6, 5, 6, -1, 4, 7, -2, 0, -2, 3, 2, -1, 2, 1, 5, 1], [1, -7, -1, 1, 1, 0, -1, -2, 2, 2, -1, -3, -3, 2, 1, 3, -1, -7, -3, 1], [-8, 1, 0, -2, 6, -6, 2, 1, -1, -1, 2, 3, 3, 0, 0, 0, -2, 2, 1, 0], [-2, 1, -3, 0, 0, -2, 0, 0, -1, 2, 1, -2, 0, -4, -1, -1, 0, -2, -1, 0], [5, 0, -2, 0, 2, -3, 4, 0, -1, 0, 2, 1, 1, -1, 2, 1, -4, -3, -2, -1], [0, -4, 4, 0, -1, -1, 3, -2, 4, 0, -8, 0, 0, 1, 4, 3, 0, 0, 0, 0], [-7, 1, -1, -3, -3, 0, -1, 0, -1, -6, 5, 3, 1, 0, -1, -1, 2, 3, 3, -1], [-5, -1, 0, 2, 7, -6, 0, -1, -2, 1, -8, -2, -2, 1, 0, 1, 0, -3, -1, 1], [-5, 3, 2, -1, -1, 4, 3, 1, -1, 3, 4, 1, 2, -4, -4, -4, 0, 0, 2, -1], [-4, -2, 4, -2, -6, -2, -6, -2, 0, 0, 0, 2, 6, -2, -4, -2, -6, -4, 2, -4], [2, -2, -2, -2, 0, 4, 4, 2, -5, -4, 2, 0, 4, -4, -2, -2, -1, 0, 0, 0], [4, 1, 4, -2, -2, -8, 4, -3, -1, -3, 2, -3, -1, 2, 2, 2, -2, 0, -1, -2], [8, 2, -2, 2, 2, 0, 2, 0, 0, 10, -2, 0, 2, 0, -2, -2, -2, -4, -4, -2], [-1, -1, -1, 3, -1, 0, 1, 0, 0, 4, 3, 5, -3, -2, -3, -1, 3, -1, -1, -1], [0, -4, -2, 0, -2, -2, 6, -2, -2, -4, -2, -6, 6, -2, -2, 0, 0, 0, 0, 0], [9, -1, -3, 1, 3, 0, -3, 4, -1, 4, 3, 5, -5, 0, 5, -3, 4, -3, -3, 1], [-2, -3, -3, -1, 3, -2, -5, 4, 2, 1, -1, -2, 1, 2, -1, -1, 1, 2, 2, 3], [-5, 0, 3, 1, -7, 8, 3, -3, 2, 3, -1, 2, 2, 2, -3, -1, -2, -1, 0, 1], [-4, 0, 10, -4, -4, -8, 4, -6, 0, 6, 2, 0, 6, 0, -2, -2, -4, -2, 2, -4], [1, 3, 4, -8, 2, 0, -8, -1, -8, 4, 0, 3, 9, 0, 2, 4, -7, -4, 1, -2], [5, 4, 0, 0, -2, 5, -4, 2, -1, -2, 0, 3, 3, -1, 0, -1, 2, 1, 2, 3], [2, 0, -4, 4, 0, -2, 2, 0, 2, 0, 0, 2, -6, 0, 0, 0, 4, 2, 0, 2], [2, 4, -2, 2, 4, -6, 1, 2, -2, -2, -2, -4, -4, -3, 2, -1, 2, 2, 0, -2], [-3, 3, -4, 3, 3, 8, -5, 3, 5, 3, 2, -3, -8, 4, 2, 2, 2, 2, 0, 3], [0, 4, 5, -3, 1, 6, 3, 3, 3, -2, 1, -5, -2, 4, -3, -3, 3, 4, 5, 1], [-2, 1, -2, 2, -1, 1, -5, 1, 4, -1, 8, -3, -3, -1, 2, 1, 3, -2, -1, 0], [8, 8, 0, 0, -4, -2, 0, 0, -2, -12, 0, 0, -4, -8, 0, 0, 4, 0, -4, -4], [-8, -1, -3, -2, 6, 0, -2, 0, -2, -2, -1, 7, 2, 0, -3, -3, -3, -5, -2, -6], [-4, 0, -2, 0, 2, 0, 2, 2, 1, -2, 4, 0, 0, 2, 2, 4, -3, -2, 0, 2], [-2, -10, 4, 4, -2, 0, -4, -6, -4, -6, 4, 4, 0, -6, 0, 6, -2, -2, -6, -2], [-5, 2, 4, 1, -5, 0, -3, -2, -1, -8, -4, -1, 1, 0, 0, 4, -1, -1, -4, 5], [7, -1, 2, -2, -6, -8, -2, -7, -4, -6, -2, 1, 3, -6, 4, 4, -3, -4, -7, -6], [-6, 4, 0, 0, -4, 2, 2, 0, -2, 0, 0, 2, 2, -4, 0, -4, 0, 2, 0, -2], [-7, -4, 1, -1, 7, -8, -7, -5, -2, 5, -3, 6, 6, 2, -1, 1, -2, -7, -4, -1], [-1, -4, 2, 0, 5, 6, 4, -2, -3, -4, 4, -1, 3, -1, 0, 1, 0, -1, 0, -1], [1, -2, 7, -5, -5, -6, 5, -5, -5, -1, 3, 0, 6, -2, 1, 1, -5, -1, -2, -5], [8, -2, 2, -2, 8, -6, 2, 0, 0, 0, -2, 2, 2, 4, -2, 2, -2, 2, 2, 2], [1, -6, -5, 5, -1, -2, 1, 3, 2, 15, -5, -2, 0, 0, -3, -3, 2, 1, 0, 1], [-5, 3, -1, 1, 1, 0, -1, 2, -1, -10, 5, -3, -1, 0, 3, -1, 4, 3, 1, 3], [8, -4, 0, -2, 10, -6, 7, 4, 6, -4, 8, -4, -4, -3, 0, -3, 2, 4, 4, 4], [-6, 4, -4, -2, 2, 0, 2, 0, 0, -4, 4, 8, 2, 0, -4, -4, 2, 0, -2, -2], [-3, -1, 1, -1, 5, -4, -3, 2, 2, -2, 1, -3, -3, 4, -1, 3, 3, 5, 3, 1], [5, 6, -7, 3, 3, -2, 2, 9, 3, -1, 1, 0, -10, 7, 3, -4, 7, 5, 2, 5], [10, -1, -1, 0, 0, 6, -4, 0, 1, -10, -5, 0, 0, 0, 1, 1, 2, 0, -1, 0], [4, 2, -3, -5, -11, 0, 1, 1, 0, 4, 11, 5, 0, 0, -1, -3, 6, 8, 7, -1], [-2, 0, -2, 2, -5, -3, -3, 0, -2, -6, -2, 4, 0, -7, -6, 1, 2, -2, -8, -2], [6, -2, -4, 4, 2, 0, 8, 2, 8, 0, 4, 2, -10, 0, 0, -4, 10, 8, 2, 4], [4, -6, -2, 4, 2, -2, -4, 0, 2, 6, -2, -4, -4, 4, 6, 2, -2, 4, 2, 4], [-9, -6, 3, -1, -1, -8, 1, -1, -3, 9, 3, 6, 6, 2, 1, 1, -5, 1, 0, -1], [7, -1, -2, 0, -2, 0, -4, 1, -2, 8, 6, 3, -1, 0, 0, -2, 3, -2, -1, -2], [3, -4, -3, 3, 11, 0, 7, 5, 4, -1, 1, -4, -6, 0, 7, -3, 2, -1, 2, -1], [0, -3, 4, 0, 10, 0, -2, 1, -1, -3, 2, -1, -1, 0, 6, 4, 0, -6, -3, 2], [2, 4, 2, -6, 6, 0, -6, -2, -6, 6, 2, 2, 8, 0, -2, 2, -4, 0, 2, -2], [0, 2, -4, -2, 16, 16, 4, 2, -1, -8, 2, -6, 6, 2, -4, -6, -1, 0, 6, 0], [0, -3, 9, -5, -5, 2, -1, -4, -2, -1, -1, 4, 9, -2, -1, -1, -3, 0, 0, -5], [7, 6, -3, 1, 7, -14, 1, -3, -8, 3, -3, 6, 6, 0, 3, 3, -8, -5, -6, -7], [-5, 8, 1, 5, 5, 2, 3, 7, 7, 3, -3, 0, -12, 10, -1, -1, 9, 7, 6, 5], [-7, 6, 0, 2, 0, -1, 2, 4, 1, -4, 0, -7, -7, -3, 0, -3, 6, 9, 4, -1], [-4, 0, 10, -6, -14, -10, -4, -6, -4, -2, 2, 2, 6, -4, 2, 6, -6, -4, -4, -4], [2, 6, -2, -2, -2, -4, -2, 0, 2, -12, -6, -2, -2, 0, 2, 2, 0, 2, -2, 2], [0, -12, 0, 0, -4, 8, 3, -4, 2, 4, -4, 0, 0, 5, 0, 5, -6, -12, -4, 4], [5, 7, -6, 7, -1, -8, -5, -1, 2, -5, 4, 5, -6, -8, 0, 0, 1, -4, -4, -1], [1, -1, 0, 0, 8, 7, 0, 5, 0, 13, 2, -2, 0, 5, 2, -3, 0, 1, 3, 1], [8, 3, -5, -1, 3, -10, 3, 4, 0, -9, 5, -8, -3, -2, -3, -3, 1, 0, 2, 3], [3, 4, -4, -2, 13, 10, 0, 4, 1, 10, 2, -7, 1, 7, 2, -5, -2, 3, 8, 3], [6, -2, -2, -2, -4, 0, 6, 0, 0, 0, 6, 0, 0, -10, 2, -8, 0, 0, -2, -4], [7, 0, 1, 5, 9, 0, -3, -1, 1, 3, 1, -6, -4, 0, -1, 1, 5, 5, 0, 5], [-2, 2, -2, 2, 7, 9, 1, 6, 2, -12, 2, 6, -2, 1, -6, -7, 4, -2, 2, -2], [-9, 0, -3, -1, 13, 0, 1, 1, -2, -3, -7, 0, 4, 0, -1, -3, -8, -5, 0, -7], [-5, -10, 6, 5, -1, 0, -3, -4, 1, 2, -2, -9, -1, 0, -2, 6, -3, -7, -2, -3], [-7, 1, -2, 2, -2, 8, -10, -1, 2, 6, -6, 5, 1, -6, 0, 0, 3, -2, -3, -2], [11, -4, -1, -3, -11, 8, -5, -3, -2, 3, 3, 6, 6, -2, 1, -1, -2, -5, -4, -3], [-1, 4, -7, 1, 1, -4, 3, 5, -4, -3, 5, 0, 0, -10, -5, -5, 2, -1, 0, 1], [8, 4, 2, 0, 4, 12, 4, -2, 8, -2, -2, -12, -6, 8, -2, -2, 2, -2, 6, 4], [10, -3, 5, -1, -13, 0, -1, 0, -1, 9, -3, -2, 1, 0, 5, 5, -4, -6, -2, 1], [0, 6, 2, -6, 4, 0, 2, 0, -4, 2, -2, -6, 10, 0, 2, 2, -8, 2, 6, -2], [9, -7, -2, -2, 2, 16, 6, -5, -2, -2, -6, -9, 1, 6, 4, 4, -3, -4, -1, 2], [8, -4, -6, 2, -4, 8, -6, 2, -2, -2, -2, -2, -2, -2, 6, 4, 0, -12, -4, 4], [1, -4, 1, 1, -9, -10, 0, -1, -5, -3, 1, -2, 0, -3, 3, 6, 1, 1, -8, 1], [-2, -1, -1, -4, -14, 0, 2, 0, -1, 4, 9, 3, 4, 0, -1, -1, 0, -3, 2, -6], [6, -8, -5, 9, 7, 0, 3, 1, 8, 4, -3, 3, -14, 0, -3, -5, 8, 2, -3, 1], [-3, -9, 5, -3, 4, 7, -10, -4, -3, 10, -5, -7, 13, -3, -1, 2, -12, -3, -1, -3], [-3, -6, -2, -4, 1, 0, 6, 4, 4, -4, 10, -3, -3, -1, 2, 1, -1, 3, 2, 1], [-9, 3, 12, -1, -1, 20, 3, -5, 7, 13, -2, 1, 4, 4, -6, -6, 0, -2, 8, -1], [0, 8, -6, 0, 10, 10, 4, 2, 2, -8, 2, 2, -2, 0, -6, -6, 4, 0, 4, 0], [8, -5, 6, 4, 10, -6, 8, -3, 11, 3, 0, -9, -9, 2, -6, -4, 4, 2, 3, 4], [7, -5, 1, -5, 3, -10, -1, 4, 1, -8, 7, -5, 5, 4, -3, -3, -4, 3, 5, 3], [-5, 0, 3, -5, -9, 0, -1, 5, -3, -13, 7, 0, 0, 0, 13, 3, 1, -3, 2, 3], [-5, 1, 5, -1, 1, 0, -1, -2, -1, -6, -9, -5, 3, 0, 1, 5, -6, 1, 1, 3], [-2, 0, 6, -4, -14, -12, 4, -6, -2, 12, 2, 0, 4, -2, 2, 4, -2, -2, 0, -2], [12, 2, -1, 3, -1, -14, -3, -1, 1, -12, 3, 3, -4, 0, 3, 3, -1, 0, -3, -1], [-9, -3, -1, -1, -3, 2, 1, 0, -7, 6, 5, 9, 11, -12, -5, -5, -4, -3, -1, -3], [3, 2, -1, -1, 3, 0, 0, -5, 3, -7, -9, -4, -2, -1, 5, 6, -1, 3, -2, 3], [8, 2, -6, 4, 0, -4, -8, 4, 0, -12, -2, 8, 0, -8, -2, -2, 4, 0, -2, 0], [3, 3, 2, -4, 18, 0, -4, -3, -4, 8, -2, 5, 7, 0, -4, 2, -5, -4, -3, -2], [4, 8, -6, -2, 14, 10, -4, 2, -6, 2, 6, -6, -2, 0, 2, 2, 0, 4, -4, 4], [-9, 5, -2, 5, -3, -4, 9, 1, -8, 5, 4, 11, -2, -8, 0, 0, 1, 0, -4, -3], [7, -3, -12, 6, 0, -7, 2, 7, 0, -13, -2, -10, -4, 5, 2, -3, 4, 7, 1, 7], [-2, 6, -7, 1, 1, 10, -5, 5, 1, -2, -3, -5, -8, -4, 1, 1, 7, 2, -3, 1], [-7, 2, -5, 1, 10, -7, 4, 5, -5, -5, -5, -2, -2, -1, 5, 4, -3, -3, 0, 3], [-6, 3, -3, -2, 4, 0, 4, 2, 1, -8, 3, -1, 0, 0, 1, -3, 2, 7, 6, 0], [-16, 4, 6, -2, 6, -4, -8, 2, 3, -2, 0, 0, 0, 8, -6, 2, 7, 14, 8, 2], [11, -2, -3, 5, -7, 0, 1, 5, 1, 5, 1, -8, -6, 0, 7, -3, 3, 1, 2, 1], [4, 8, -4, -4, 7, 3, 7, 2, -2, 12, 4, -8, 0, 5, 4, -1, 2, 4, 4, 4], [-7, 2, 9, -3, -9, -16, -7, -5, -2, 3, 9, 8, 4, -6, 3, 3, -6, 1, -6, -9], [-12, -2, 4, 0, -2, 6, -2, -2, 0, 2, -8, 2, 2, 10, -4, 6, -2, 0, 2, 4], [1, -1, -6, 4, 5, 4, 0, 9, 1, -1, -4, 0, -2, 1, -4, -5, 5, 1, -3, 1], [4, 2, 0, -6, 12, -18, -2, 2, -4, -2, 4, 4, 4, 0, 0, 0, 0, 10, 2, -6], [2, 10, 2, -4, -2, -4, 6, -4, 2, -2, -2, 2, -6, 0, 6, 6, 4, 2, -2, 2], [4, 4, -8, 4, 10, 0, 4, 4, 4, -4, 0, 4, -8, 0, 0, -8, 8, 8, 0, 4], [7, -8, 3, 5, 3, 0, 9, -3, 11, 5, -5, -2, -10, 0, -3, 3, 5, 7, 2, 5], [-7, 7, 14, -12, -2, 5, 4, -5, 0, -5, 4, 4, 10, 1, 0, -1, -6, -7, 7, -7], [-1, -2, 3, -5, -9, -8, -12, -5, -7, 5, -1, -8, 10, -5, 1, 6, -11, -1, -6, -1], [-4, 7, -7, 6, -2, -14, 2, -2, -3, 4, 5, 2, -8, -8, 3, 3, 0, -4, -7, -2], [-9, -8, 0, -4, 0, 1, 0, -2, -1, 2, 0, 5, 5, -7, 0, -7, 0, -5, -2, 1], [7, -5, -1, 3, -2, -9, 4, 2, 3, -12, -3, -7, -7, 9, 13, 4, 2, 7, 3, 7], [13, -5, 0, 2, 5, -6, 4, -5, 0, 5, -4, 0, 0, -1, 0, -1, -4, -9, -5, -1], [7, -1, 0, -1, -1, -8, -1, 5, -4, -13, -2, 5, 0, 0, 2, 2, 3, 6, -2, -1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_240_s_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_240_2_s_c();".
// 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_s_c(:prec:=20)
    chi := MakeCharacter_240_s();
    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_s_c();".
// 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_s_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_240_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![262144, 0, 76283904, 0, 204587008, 0, 145248256, 0, 49590528, 0, 9697664, 0, 1161104, 0, 86368, 0, 3880, 0, 96, 0, 1]>],Snew);
    return Vf;
end function;