
// Make newform 605.2.j.i in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_605_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_605_j_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_605_2_j_i();".
// 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_605_2_j_i();".
// 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 := [6561, -2187, 2187, -1944, 648, 189, 54, 168, -137, 56, 6, 7, 8, -8, 3, -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, 0], [32805, -1444392, 10935, -9720, 3240, 945, -90557, 840, -685, 280, 30, -5709, 40, -40, 15, -5], [-524880, -349920, 602640, -19440, 103680, -66960, -19440, 39680, -2480, 6480, -4960, -1040, 3467, 0, 400, -160], [13122, -10935, 0, -8920, 1053, 1674, -729, 93, -496, 81, 93, -48, 3, -31, 6, 3], [-19683, 19683, -17496, 5832, -20557, 486, 1512, -1233, 504, -664, 63, 72, -72, 27, -40, 9], [146286, 0, 0, 0, 0, 22258, 0, 0, 0, 0, 718, 0, 0, 0, 0, 31], [203391, -301320, 67797, -60264, 20088, 5859, -20584, 5208, -4247, 1736, 186, -501, 248, -248, 93, -31], [-1762722, 1468935, 0, 266409, -141453, -224874, 97929, -12493, 28336, -10881, -12493, 6448, -403, 1053, -806, -403], [3682908, -3069090, 0, 5265, 295542, 469836, -204606, 26102, 4150, 22734, 26102, -13472, 842, -324, 1684, 842], [-4719546, 3932955, 0, -1310985, -378729, -602082, 262197, -33449, -79846, -29133, -33449, 17264, -1079, -6561, -2158, -1079], [-6561, 386, -2187, 1944, -648, -189, -54, -168, 137, -56, -6, -7, -8, 8, -3, 1], [2119203, 1412802, -164997, 78489, -418608, 270351, 78489, 14266, 10013, -26163, 20026, 4199, 178, 0, -1615, 646], [-6561, -4374, -3955, -243, 1296, -837, -243, -222, -31, 81, -62, -13, -15, 0, 5, -2], [1310013, -1310013, 1164456, -388152, -113211, -32346, -100632, 82063, -33544, -3594, -4193, -4792, 4792, -1797, -130, -599], [7118685, 4745790, -2066715, 263655, -1406160, 908145, 263655, -189212, 33635, -87885, 67270, 14105, -8855, 0, -5425, 2170], [1627128, -1014768, 542376, -482112, 160704, 46872, -77435, 41664, -33976, 13888, 1488, -4008, 1984, -1984, 744, -248]];
    Rf_basisdens := [1, 261711, 785133, 9693, 58158, 87237, 174474, 523422, 785133, 1570266, 2154, 785133, 2154, 523422, 1570266, 261711];
    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_605_j();"
function MakeCharacter_605_j()
    N := 605;
    order := 10;
    char_gens := [122, 486];
    v := [5, 6];
    // 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_605_j_Hecke();"
function MakeCharacter_605_j_Hecke(Kf)
    N := 605;
    order := 10;
    char_gens := [122, 486];
    char_values := [[-1, 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]];
    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, 1, 1, 0, 0, -1, 0, 0, 1, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, -2, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, -2, 0, 0, 0, 0, 0], [0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, -1, 1, 0, -1, -1, 0, 1, 1, 1, 0], [0, 0, 0, -2, 0, 0, 0, -2, 0, -2, 0, 0, 0, 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, 3, 2, -3, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -2, 0], [0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -4, 0, -4, 0], [0, 4, 0, 0, 4, 4, 0, 0, -4, 0, 0, 4, 0, -4, 0, 0], [0, 0, 0, 4, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, -2, 0, 0, 0, 0, -6], [0, 0, 0, 0, 3, 4, 3, -3, 0, 3, 3, 0, -3, -3, -3, 0], [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0], [8, 0, -8, 8, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -8], [0, -2, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 0, 0], [2, 0, 0, 0, 1, 0, -1, 1, 0, 1, 1, 0, 1, 1, -1, 0], [0, -2, 0, 0, -3, -2, 0, 0, 2, 0, 0, -2, 0, 3, 0, 0], [-6, 0, 6, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6], [0, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -4, 0, -4, 0], [-10, 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, -2, 5, 0, 5, 0], [0, -2, 0, 0, 0, 0, 6, 0, 0, 0, 6, 0, 0, 0, 0, 0], [4, 0, 0, 0, 2, 0, -2, 2, 0, 2, 2, 0, 2, 2, -2, 0], [0, 6, 0, 0, 0, 0, -5, 0, 0, 0, -5, 0, 0, 0, 0, 0], [0, 0, 0, -4, 0, 0, 0, 6, 0, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 4, 0, 0, 0, 0, 2], [0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0, 6, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, -9, 0, -9, 0], [0, -2, 0, 0, 0, -2, 0, 0, 2, 0, 0, -2, 0, 0, 0, 0], [0, 10, 0, 0, -2, 10, 0, 0, -10, 0, 0, 10, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0, 2, 0], [0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 5, 0], [0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 3, 0, 0, 0, 0, -14], [0, 0, 0, -16, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, -12, 0, 0, 0, -12, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 4, 2, -2, 0, 2, 2, 0, -2, -2, -2, 0], [8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [12, 0, -12, 12, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, -12], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 3, 0], [0, 0, 0, 0, 0, 0, 0, 8, -6, -8, 0, 0, 0, 0, 0, 0], [18, 0, -18, 18, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -18], [0, -4, 0, 0, 4, -4, 0, 0, 4, 0, 0, -4, 0, -4, 0, 0], [0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -2, 0], [10, 0, 0, 0, -2, 0, 2, -2, 0, -2, -2, 0, -2, -2, 2, 0], [12, 0, -12, 12, -3, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, -12], [0, 0, 0, 0, 0, 0, 0, -4, -8, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, -14, -4, 4, 0, -4, -4, 0, 4, 4, 4, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 4, 0, 0, 0, 0, 2], [0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 6, 4, 0, 0, -4, 0, 0, 4, 0, -6, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 4, 0, 0, 0, 0, 14], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, -6, 0, -6, 0], [0, 0, 0, 0, 0, 0, 0, -4, 0, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 18, 10, 18, -18, 0, 18, 18, 0, -18, -18, -18, 0], [0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, -4, 0], [0, 14, 0, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0, 0, 0], [0, -14, 0, 0, -11, -14, 0, 0, 14, 0, 0, -14, 0, 11, 0, 0], [-4, 0, 0, 0, 3, 0, -3, 3, 0, 3, 3, 0, 3, 3, -3, 0], [0, 0, 0, 0, 0, 0, 0, -6, 16, 6, 0, 0, 0, 0, 0, 0], [0, -14, 0, 0, 0, 0, -2, 0, 0, 0, -2, 0, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 4, 0], [0, 0, 0, 0, -1, -14, -1, 1, 0, -1, -1, 0, 1, 1, 1, 0], [0, 0, 0, -16, 0, 0, 0, -4, 0, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 15, -8, -15, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 6, 4, 6, -6, 0, 6, 6, 0, -6, -6, -6, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, -4], [0, -4, 0, 0, -5, -4, 0, 0, 4, 0, 0, -4, 0, 5, 0, 0], [0, 0, 0, -2, 0, 0, 0, -5, 0, -5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -12, -4, -12, 12, 0, -12, -12, 0, 12, 12, 12, 0], [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, -4, 0, 0, 0, 0, -2], [14, 0, -14, 14, -4, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, -14], [-4, 0, 0, 0, -8, 0, 8, -8, 0, -8, -8, 0, -8, -8, 8, 0], [0, 0, -22, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 4, 0], [-8, 0, 8, -8, -10, 0, 0, 0, 0, 0, 0, 0, 0, -10, 0, 8], [0, 0, 0, 0, 0, 0, 0, -15, 10, 15, 0, 0, 0, 0, 0, 0], [-8, 0, 0, 0, 4, 0, -4, 4, 0, 4, 4, 0, 4, 4, -4, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 5, 0, 5, 0], [-10, 0, 10, -10, -5, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 10], [0, -12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-18, 0, 0, 0, -6, 0, 6, -6, 0, -6, -6, 0, -6, -6, 6, 0], [0, 0, 0, 0, 3, 16, 3, -3, 0, 3, 3, 0, -3, -3, -3, 0], [0, -4, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 4, 0, 0, 0, 0, 8], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -9, 0, -9, 0], [0, 0, 0, 20, 0, 0, 0, -4, 0, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, -4, 0, -4, 0], [0, 0, 26, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0], [0, 0, 0, 2, 0, 0, 0, -7, 0, -7, 0, 0, 0, 0, 0, 0], [0, -14, 0, 0, 0, 0, 6, 0, 0, 0, 6, 0, 0, 0, 0, 0], [-14, 0, 14, -14, 6, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 14], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -14, -6, 0, -6, 0], [0, 0, 0, 0, 0, 0, 0, -10, -4, 10, 0, 0, 0, 0, 0, 0], [0, -10, 0, 0, -2, -10, 0, 0, 10, 0, 0, -10, 0, 2, 0, 0], [0, 0, -34, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 2, 0], [-12, 0, 0, 0, -4, 0, 4, -4, 0, -4, -4, 0, -4, -4, 4, 0], [0, 14, 0, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 22, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 20, 4, 20, -20, 0, 20, 20, 0, -20, -20, -20, 0], [0, 0, 0, 0, 0, 0, 8, 0, 0, 0, -8, 0, 0, 0, 0, -16], [0, 0, 0, 2, 0, 0, 0, -12, 0, -12, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, -2, 0, 0, 2, 0, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, -12, 0, -12, 0], [0, 0, 0, 0, -4, -8, -4, 4, 0, -4, -4, 0, 4, 4, 4, 0], [0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 3, 0], [0, 0, 0, 0, 0, 0, 0, 7, 0, 7, 0, 0, 0, 0, 0, 0], [0, 0, 22, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0], [0, 8, 0, 0, 0, 0, -21, 0, 0, 0, -21, 0, 0, 0, 0, 0], [0, 4, 0, 0, 19, 4, 0, 0, -4, 0, 0, 4, 0, -19, 0, 0], [0, 0, 0, 0, 0, 0, 0, 17, -10, -17, 0, 0, 0, 0, 0, 0], [-28, 0, 0, 0, -2, 0, 2, -2, 0, -2, -2, 0, -2, -2, 2, 0], [14, 0, 0, 0, 9, 0, -9, 9, 0, 9, 9, 0, 9, 9, -9, 0], [0, 8, 0, 0, 6, 8, 0, 0, -8, 0, 0, 8, 0, -6, 0, 0], [0, 28, 0, 0, 0, 0, -2, 0, 0, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 8, -14, 8, -8, 0, 8, 8, 0, -8, -8, -8, 0], [-28, 0, 28, -28, 5, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 28], [0, 0, 26, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, -4, 0], [0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 3, 0, 0, 0, 0, 34], [0, 0, 0, 0, 0, 0, 0, 9, 0, 9, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 9, 4, 9, -9, 0, 9, 9, 0, -9, -9, -9, 0], [0, 0, 0, 0, 0, 0, 0, 12, 0, -12, 0, 0, 0, 0, 0, 0], [-4, 0, 4, -4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4], [0, -2, 0, 0, 0, 0, 10, 0, 0, 0, 10, 0, 0, 0, 0, 0], [0, 0, -24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0], [0, -12, 0, 0, -24, -12, 0, 0, 12, 0, 0, -12, 0, 24, 0, 0], [-26, 0, 26, -26, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 26], [18, 0, 0, 0, 14, 0, -14, 14, 0, 14, 14, 0, 14, 14, -14, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, -4, 0, -4, 0], [0, -6, 0, 0, 0, 0, 6, 0, 0, 0, 6, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, -11, 0, 0, 0, -11, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, 0, 0, -26], [0, 0, -20, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0, 6, 0], [0, 0, 0, -18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 20, 0, 0, 27, 20, 0, 0, -20, 0, 0, 20, 0, -27, 0, 0], [0, -10, 0, 0, -14, -10, 0, 0, 10, 0, 0, -10, 0, 14, 0, 0], [0, 0, 0, -10, 0, 0, 0, 9, 0, 9, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, -3, 0], [0, -12, 0, 0, 0, 0, 18, 0, 0, 0, 18, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, -12, -4, 4, 0, -4, -4, 0, 4, 4, 4, 0], [-4, 0, 0, 0, -3, 0, 3, -3, 0, -3, -3, 0, -3, -3, 3, 0], [0, -2, 0, 0, 0, 0, -8, 0, 0, 0, -8, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [10, 0, 0, 0, 5, 0, -5, 5, 0, 5, 5, 0, 5, 5, -5, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -14, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 14, 4, 0, 0, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, 12, 0, 0, 0, 12, 0, 0, 0, 0, 0], [40, 0, -40, 40, -4, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, -40], [0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 6, 0, 0, 0, 0, -8], [0, 0, 0, 0, 0, 0, 8, 0, 0, 0, -8, 0, 0, 0, 0, -34], [0, -20, 0, 0, -24, -20, 0, 0, 20, 0, 0, -20, 0, 24, 0, 0], [0, 0, 0, 0, 0, 0, 12, 0, 0, 0, -12, 0, 0, 0, 0, 18], [0, 0, 0, 0, -25, -4, -25, 25, 0, -25, -25, 0, 25, 25, 25, 0], [0, 0, 0, 0, 0, 0, 0, -22, 8, 22, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 6, -18, 6, -6, 0, 6, 6, 0, -6, -6, -6, 0], [0, 0, -28, 0, 0, 0, 0, 0, 0, 0, 0, 0, -11, 0, 11, 0], [0, 26, 0, 0, 7, 26, 0, 0, -26, 0, 0, 26, 0, -7, 0, 0], [0, 0, 0, 0, 0, 0, 0, 5, -12, -5, 0, 0, 0, 0, 0, 0], [-8, 0, 0, 0, -8, 0, 8, -8, 0, -8, -8, 0, -8, -8, 8, 0], [0, 0, 0, 0, 0, 0, 0, -6, -4, 6, 0, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_605_j_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_605_2_j_i();".
// 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_605_2_j_i(:prec:=16)
    chi := MakeCharacter_605_j();
    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_605_2_j_i();".
// 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_605_2_j_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_605_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![256, 0, -448, 0, 720, 0, -1148, 0, 1829, 0, -287, 0, 45, 0, -7, 0, 1]>,<19,R![256, -64, 16, -4, 1]>],Snew);
    return Vf;
end function;