
// Make newform 600.2.b.i in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_600_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_600_b_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_600_2_b_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_600_2_b_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 := [4, 0, 64, 0, 352, 0, 880, 0, 1092, 0, 672, 0, 192, 0, 24, 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, 44, 0, 594, 0, 1638, 0, 1488, 0, 506, 0, 69, 0, 3, 0, 0], [-22, 0, -142, 0, -584, 0, -962, 0, -651, 0, -191, 0, -24, 0, -1, 0], [-16, 0, -596, 0, -3024, 0, -5334, 0, -3984, 0, -1250, 0, -164, 0, -7, 0], [-28, -56, -212, -282, 20, 218, 1352, 1922, 1674, 2176, 630, 799, 90, 113, 4, 5], [76, 28, 1496, 594, 6808, 1638, 11726, 1488, 8638, 506, 2692, 69, 352, 3, 15, 0], [-76, 28, -1496, 594, -6808, 1638, -11726, 1488, -8638, 506, -2692, 69, -352, 3, -15, 0], [-300, 484, -3428, 3014, -12164, 4770, -18152, 1686, -12334, -1074, -3698, -657, -474, -107, -20, -5], [-236, -272, -2440, -2004, -8004, -6544, -11972, -9702, -8350, -6672, -2548, -2018, -330, -260, -14, -11], [72, 344, 664, 3498, 1440, 11966, 918, 17582, 4, 11832, -100, 3529, -20, 451, -1, 19], [-72, 344, -664, 3498, -1440, 11966, -918, 17582, -4, 11832, 100, 3529, 20, 451, 1, 19], [-268, 272, -3632, 2004, -14052, 6544, -22640, 9702, -16318, 6672, -5048, 2018, -658, 260, -28, 11], [0, -252, 0, -2412, 0, -7410, 0, -10334, 0, -6862, 0, -2042, 0, -261, 0, -11], [-136, -38, -1608, -1046, -6092, -4916, -9752, -8530, -7004, -6311, -2164, -1971, -282, -258, -12, -11], [136, -38, 1608, -1046, 6092, -4916, 9752, -8530, 7004, -6311, 2164, -1971, 282, -258, 12, -11], [0, 540, 0, 5842, 0, 20960, 0, 32332, 0, 22674, 0, 6921, 0, 896, 0, 38]];
    Rf_basisdens := [1, 8, 4, 8, 16, 16, 16, 16, 16, 16, 16, 16, 8, 8, 8, 8];
    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_600_b();"
function MakeCharacter_600_b()
    N := 600;
    order := 2;
    char_gens := [151, 301, 401, 577];
    v := [1, 1, 1, 2];
    // 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_600_b_Hecke();"
function MakeCharacter_600_b_Hecke(Kf)
    N := 600;
    order := 2;
    char_gens := [151, 301, 401, 577];
    char_values := [[-1, 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], [-1, 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]];
    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, 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, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, -1, 0, 0, 1, 1, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0], [0, 0, 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, -1, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -1, -1, 0], [0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, -2], [0, 0, 0, 0, -2, 0, 0, 2, 0, 0, 0, 0, 0, -1, 1, 0], [0, 0, 0, 0, 0, 1, -1, 0, 2, 0, 0, 2, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 3, 0, 0, 0, 0, 0, -3, 3, 0, 0, 0, 0, 0], [0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 0], [0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 0, 0, 4, -2, -2, 0], [0, 0, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 3, 3, 0], [0, 0, 0, 0, 0, 1, 1, 0, 0, 4, 4, 0, 0, 0, 0, 0], [0, 2, 0, 0, -2, 0, 0, 2, 0, 0, 0, 0, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0], [0, 0, -1, 0, -5, 0, 0, -5, 0, 0, 0, 0, -5, 2, 3, 0], [0, 0, 0, -2, 0, 0, 0, 0, -2, 0, 0, 2, 0, 0, 0, -1], [0, 0, 0, 0, 0, -3, -3, 0, 0, 2, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 3, 3, 0], [0, 0, 0, 0, 0, 1, 1, 0, 0, -3, -3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 5, 0, 0, 3, 0, 0, -3, 0, 0, 0, 0, 0, -1, -4, 0], [0, 0, 0, 0, 0, -1, -1, 0, 0, -5, -5, 0, 0, 0, 0, 0], [-6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -3, 0], [0, 0, 3, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0], [0, 0, 0, 3, 0, 0, 0, 0, 3, 0, 0, -3, 0, 0, 0, 6], [0, 0, 0, 0, 0, -1, 1, 0, -2, 0, 0, -2, 0, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0, 0, 0, 4, -4, 0, 0, 0, 0, 0], [0, 0, 0, -7, 0, 0, 0, 0, 0, -3, 3, 0, 0, 0, 0, 0], [0, 3, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, -1, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 2, 2, 0], [0, 0, 2, 0, -2, 0, 0, -2, 0, 0, 0, 0, -2, 2, 0, 0], [0, 0, 0, 0, 0, -6, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -5, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 1, 1, 0], [10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 3, 0], [0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, -4, 0, 0, 0, -1], [0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 4, 4, 0], [0, 0, 0, 0, 0, -7, -7, 0, 0, -9, -9, 0, 0, 0, 0, 0], [0, 0, -6, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2, -4, 2, 0], [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -3, 0], [0, -9, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 5, 4, 0], [0, 0, 0, 0, 0, 3, 3, 0, 0, -5, -5, 0, 0, 0, 0, 0], [0, 0, 0, -11, 0, 0, 0, 0, 0, 6, -6, 0, 0, 0, 0, 0], [0, 0, -1, 0, -3, 0, 0, -3, 0, 0, 0, 0, -3, 1, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 5, 0], [0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, -6], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0], [0, 0, 0, 0, 0, -3, 3, 0, -6, 0, 0, -6, 0, 0, 0, 0], [0, 0, 0, 3, 0, 0, 0, 0, 0, -5, 5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3, 3, 0, 2, 0, 0, 2, 0, 0, 0, 0], [0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 4, 0], [0, 0, 0, 0, 0, 6, -6, 0, -2, 0, 0, -2, 0, 0, 0, 0], [0, 0, 0, -3, 0, 0, 0, 0, 0, -5, 5, 0, 0, 0, 0, 0], [-10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -3, 0], [0, 0, 0, 0, 0, -6, 6, 0, -6, 0, 0, -6, 0, 0, 0, 0], [0, 0, 0, 0, 0, 5, 5, 0, 0, 2, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, -2, -2, 0], [0, 0, 0, 0, 0, 13, 13, 0, 0, -7, -7, 0, 0, 0, 0, 0], [0, 0, 2, 0, -10, 0, 0, -10, 0, 0, 0, 0, -10, 6, 4, 0], [0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, -4, 0, 0, 0, 5], [0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, -6], [-18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0], [0, 0, 0, 7, 0, 0, 0, 0, 0, 10, -10, 0, 0, 0, 0, 0], [0, 0, -3, 0, 3, 0, 0, 3, 0, 0, 0, 0, 3, -3, 0, 0], [0, 0, 0, -3, 0, 0, 0, 0, -3, 0, 0, 3, 0, 0, 0, 4], [0, -6, 0, 0, 6, 0, 0, -6, 0, 0, 0, 0, 0, 6, 0, 0], [10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0], [0, 1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -16, 1, 1, 0], [0, 0, 8, 0, -4, 0, 0, -4, 0, 0, 0, 0, -4, 6, -2, 0], [0, 0, 0, 0, 0, -8, 8, 0, 2, 0, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 3, 3, 0], [0, 0, 0, 0, 0, 13, 13, 0, 0, 6, 6, 0, 0, 0, 0, 0], [0, 10, 0, 0, 4, 0, 0, -4, 0, 0, 0, 0, 0, -3, -7, 0], [0, 0, 0, 0, 0, 4, -4, 0, 4, 0, 0, 4, 0, 0, 0, 0], [0, 0, -3, 0, 5, 0, 0, 5, 0, 0, 0, 0, 5, -4, -1, 0], [0, 0, 0, -2, 0, 0, 0, 0, -2, 0, 0, 2, 0, 0, 0, 5], [0, 0, 0, 0, 0, 5, 5, 0, 0, -4, -4, 0, 0, 0, 0, 0], [0, 0, 2, 0, 6, 0, 0, 6, 0, 0, 0, 0, 6, -2, -4, 0], [0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, -4, 0, 0, 0, 3], [0, 1, 0, 0, -9, 0, 0, 9, 0, 0, 0, 0, 0, -5, 4, 0], [6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -11, 11, 0], [0, 0, 0, 7, 0, 0, 0, 0, 0, -8, 8, 0, 0, 0, 0, 0], [0, 0, 1, 0, 5, 0, 0, 5, 0, 0, 0, 0, 5, -2, -3, 0], [0, 0, 0, 0, 8, 0, 0, -8, 0, 0, 0, 0, 0, 4, -4, 0], [0, 0, 0, 0, 0, 3, -3, 0, 2, 0, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, 0, 0, 0], [0, 0, 0, 0, 0, 5, -5, 0, -4, 0, 0, -4, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, 7, -7, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, -1, 0, 0, 8, 8, 0, 0, 0, 0, 0], [0, 4, 0, 0, 2, 0, 0, -2, 0, 0, 0, 0, 0, -1, -3, 0], [-10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -11, 11, 0], [0, 0, 0, 0, 0, -2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 3, 3, 0, 0, 4, 4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 13, 13, 0, 0, 5, 5, 0, 0, 0, 0, 0], [0, 0, 10, 0, -6, 0, 0, -6, 0, 0, 0, 0, -6, 8, -2, 0], [22, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -3, 0], [0, 0, 0, -6, 0, 0, 0, 0, -6, 0, 0, 6, 0, 0, 0, -7], [0, 0, 0, -5, 0, 0, 0, 0, -5, 0, 0, 5, 0, 0, 0, 0], [0, 0, 0, 0, 0, -9, -9, 0, 0, -7, -7, 0, 0, 0, 0, 0], [-6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 3, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 5, 5, 0], [0, 10, 0, 0, 4, 0, 0, -4, 0, 0, 0, 0, 0, -3, -7, 0], [0, 0, 0, 0, 0, 9, -9, 0, 4, 0, 0, 4, 0, 0, 0, 0], [0, 0, 0, 15, 0, 0, 0, 0, 0, -10, 10, 0, 0, 0, 0, 0], [0, 0, 0, 13, 0, 0, 0, 0, 0, -9, 9, 0, 0, 0, 0, 0], [0, -1, 0, 0, -3, 0, 0, 3, 0, 0, 0, 0, 0, -1, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, -13, -13, 0], [0, 0, 0, 0, 0, -12, 12, 0, -6, 0, 0, -6, 0, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0, 0, 0, -13, 13, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -9, -9, 0, 0, 2, 2, 0, 0, 0, 0, 0], [-10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0], [0, 0, -5, 0, -3, 0, 0, -3, 0, 0, 0, 0, -3, -1, 4, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, -10, -10, 0], [0, 0, -4, 0, 8, 0, 0, 8, 0, 0, 0, 0, 8, -6, -2, 0], [0, 0, 0, -8, 0, 0, 0, 0, -8, 0, 0, 8, 0, 0, 0, 3], [0, 0, 0, 5, 0, 0, 0, 0, 5, 0, 0, -5, 0, 0, 0, 10], [30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0], [0, 0, 0, -5, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -11, -11, 0], [0, 0, 0, 7, 0, 0, 0, 0, 7, 0, 0, -7, 0, 0, 0, 6], [0, 0, 0, 0, -4, 0, 0, 4, 0, 0, 0, 0, 0, -2, 2, 0], [10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -14, 14, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 15, -15, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -13, 13, 0, -4, 0, 0, -4, 0, 0, 0, 0], [0, 0, 0, -3, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0], [0, -12, 0, 0, 4, 0, 0, -4, 0, 0, 0, 0, 0, 8, 4, 0], [-6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -3, 0], [0, 0, 1, 0, 11, 0, 0, 11, 0, 0, 0, 0, 11, -5, -6, 0], [0, 0, 0, 10, 0, 0, 0, 0, 10, 0, 0, -10, 0, 0, 0, 7], [0, 0, 0, 0, 0, -1, -1, 0, 0, -14, -14, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 1, 1, 0], [0, 0, -10, 0, -6, 0, 0, -6, 0, 0, 0, 0, -6, -2, 8, 0], [0, 0, 0, -5, 0, 0, 0, 0, -5, 0, 0, 5, 0, 0, 0, 4], [0, 0, 0, 0, 0, -5, -5, 0, 0, -13, -13, 0, 0, 0, 0, 0], [-18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 5, 0], [0, 0, 0, 11, 0, 0, 0, 0, 0, -8, 8, 0, 0, 0, 0, 0], [0, 0, 0, -9, 0, 0, 0, 0, -9, 0, 0, 9, 0, 0, 0, -2], [0, 0, 0, 0, -10, 0, 0, 10, 0, 0, 0, 0, 0, -5, 5, 0], [0, 0, 0, 0, 0, -9, 9, 0, -6, 0, 0, -6, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, 14, -14, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, -1, 0, -8, 0, 0, -8, 0, 0, 0, 0], [0, 0, -4, 0, 8, 0, 0, 8, 0, 0, 0, 0, 8, -6, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, -5, -5, 0], [0, 14, 0, 0, -12, 0, 0, 12, 0, 0, 0, 0, 0, -13, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, -6, 0, 0, 0, 0], [0, 0, -5, 0, -1, 0, 0, -1, 0, 0, 0, 0, -1, -2, 3, 0], [0, 0, 0, 0, 0, 9, 9, 0, 0, 12, 12, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -19, -19, 0, 0, 9, 9, 0, 0, 0, 0, 0], [0, 0, 0, 14, 0, 0, 0, 0, 14, 0, 0, -14, 0, 0, 0, 3], [0, 15, 0, 0, -3, 0, 0, 3, 0, 0, 0, 0, 0, -9, -6, 0], [0, 0, 0, 0, 0, -9, -9, 0, 0, 9, 9, 0, 0, 0, 0, 0], [0, 0, 0, 3, 0, 0, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 7, 0], [0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_600_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_600_2_b_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_600_2_b_i(:prec:=16)
    chi := MakeCharacter_600_b();
    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_600_2_b_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_600_2_b_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_600_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![56, 0, 16, 0, 1]>,<11,R![112, 0, 24, 0, 1]>,<23,R![8, 0, -8, 0, 1]>,<43,R![2744, 0, -112, 0, 1]>],Snew);
    return Vf;
end function;