
// Make newform 4560.2.d.l in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4560_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4560_d_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_4560_2_d_l();".
// 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_4560_2_d_l();".
// 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, 63, 0, 245, 0, 362, 0, 202, 0, 35, 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], [-580, 0, -5871, 0, -9742, 0, -5482, 0, -946, 0, -27, 0], [0, 556, 0, -8645, 0, -27391, 0, -19907, 0, -3783, 0, -110], [105, 0, 2242, 0, 6550, 0, 4836, 0, 927, 0, 27, 0], [0, -2208, 0, -19380, 0, -37789, 0, -23393, 0, -4205, 0, -121], [-2220, 0, -22933, 0, -53724, 0, -36724, 0, -6864, 0, -199, 0], [0, 1052, 0, 23560, 0, 63333, 0, 44733, 0, 8437, 0, 245], [0, -1762, 0, -17955, 0, -41147, 0, -27583, 0, -5113, 0, -148], [-1304, -1358, -13946, -27569, -32256, -72224, -21634, -50682, -4008, -9542, -116, -277], [1304, -1358, 13946, -27569, 32256, -72224, 21634, -50682, 4008, -9542, 116, -277], [0, -7396, 0, -63631, 0, -136674, 0, -89894, 0, -16566, 0, -479], [-4056, 0, -61085, 0, -154190, 0, -106846, 0, -20030, 0, -581, 0]];
    Rf_basisdens := [1, 76, 76, 19, 76, 76, 76, 38, 76, 76, 76, 76];
    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_4560_d();"
function MakeCharacter_4560_d()
    N := 4560;
    order := 2;
    char_gens := [1711, 1141, 3041, 2737, 1921];
    v := [1, 2, 2, 2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4560_d_Hecke();"
function MakeCharacter_4560_d_Hecke(Kf)
    N := 4560;
    order := 2;
    char_gens := [1711, 1141, 3041, 2737, 1921];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 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, 0, 0], [1, 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, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 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, 0, 0], [0, 0, 0, 0, 1, 0, 1, -1, 1, 1, 0, 0], [0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0], [-1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 1, 0, 0, -1, 0], [0, 0, -1, 0, 1, 0, 1, -1, 1, 1, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0], [0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 1, -2, 1, 1, 0, 0], [4, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 1], [-5, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1], [-3, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0], [3, 0, 0, 1, 0, 1, 0, 0, -1, 1, 0, 0], [-2, -1, 0, 1, 0, 0, 0, 0, 1, -1, 0, -1], [0, 0, -1, 0, 1, 0, 0, 0, 0, 0, -1, 0], [0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 1, -1, 0, 0, -1, 0], [2, 1, 0, 0, 0, 1, 0, 0, -1, 1, 0, 0], [3, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [4, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, -1, -1, 1, 1, 1, 0], [0, 0, 4, 0, -1, 0, 1, 0, -1, -1, 0, 0], [4, -1, 0, 0, 0, 1, 0, 0, -1, 1, 0, 0], [0, 0, 2, 0, -3, 0, 0, 0, -1, -1, 1, 0], [-4, 1, 0, -1, 0, 0, 0, 0, 1, -1, 0, 1], [0, 0, 2, 0, 2, 0, 2, 1, 0, 0, 0, 0], [-3, 0, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0], [3, 0, 0, -1, 0, 0, 0, 0, 1, -1, 0, -2], [2, 1, 0, -1, 0, 0, 0, 0, -1, 1, 0, -1], [0, 0, 2, 0, -1, 0, 0, -3, 0, 0, 0, 0], [2, 0, 0, 1, 0, 0, 0, 0, 2, -2, 0, 1], [0, 0, -2, 0, 3, 0, 1, -1, 2, 2, 0, 0], [2, -1, 0, 1, 0, 2, 0, 0, 1, -1, 0, -1], [0, 0, -3, 0, 0, 0, -1, 0, 0, 0, 1, 0], [0, 0, 0, 0, 3, 0, -2, 0, 1, 1, -1, 0], [0, 0, 0, 0, 2, 0, -1, 1, 0, 0, -1, 0], [6, -1, 0, -3, 0, -1, 0, 0, -1, 1, 0, -1], [0, 0, 4, 0, -2, 0, 0, 0, -1, -1, 0, 0], [-2, -2, 0, -2, 0, -4, 0, 0, 2, -2, 0, 0], [1, 2, 0, 1, 0, 1, 0, 0, 1, -1, 0, 0], [0, 0, 0, -1, 0, -2, 0, 0, 2, -2, 0, 1], [-5, 0, 0, -2, 0, 1, 0, 0, -1, 1, 0, -1], [0, 0, 0, 1, 0, -1, 0, 0, 1, -1, 0, -1], [0, 0, 2, 0, 0, 0, 1, -2, 2, 2, 0, 0], [0, 0, -4, 0, 0, 0, -4, 1, -1, -1, 0, 0], [0, 0, 2, 0, 1, 0, 2, 2, -1, -1, 1, 0], [0, 0, 0, 0, -1, 0, -1, 1, 0, 0, 0, 0], [0, 0, -2, 0, -1, 0, -1, 3, 1, 1, 0, 0], [0, 0, 0, 0, -1, 0, -2, 1, 1, 1, -1, 0], [0, 0, -4, 0, 0, 0, 2, -4, 1, 1, 0, 0], [7, 0, 0, -2, 0, 0, 0, 0, -1, 1, 0, 1], [0, 0, 1, 0, -2, 0, -2, -3, 0, 0, 1, 0], [0, 0, -2, 0, -1, 0, 0, 1, 0, 0, -2, 0], [0, 0, -2, 0, -3, 0, 3, -1, 0, 0, 2, 0], [2, 0, 0, 4, 0, -2, 0, 0, -2, 2, 0, 2], [0, 0, 0, 0, 0, 0, 1, -2, 0, 0, 2, 0], [-7, 1, 0, -1, 0, -2, 0, 0, 2, -2, 0, 0], [0, 0, -4, 0, -1, 0, 3, 0, 1, 1, 0, 0], [3, 1, 0, -2, 0, 0, 0, 0, 1, -1, 0, 1], [0, 0, -2, 0, -1, 0, -4, 0, -2, -2, 0, 0], [0, 0, 3, 0, 1, 0, -2, 3, -1, -1, -1, 0], [-2, 2, 0, 0, 0, 0, 0, 0, -3, 3, 0, 4], [0, -2, 0, -1, 0, -3, 0, 0, 3, -3, 0, 1], [0, 0, -2, 0, 3, 0, 4, -2, 3, 3, 1, 0], [0, 0, 1, 0, 1, 0, 1, -1, 0, 0, -1, 0], [0, 0, 2, 0, 1, 0, 0, 2, 0, 0, -2, 0], [11, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, -1], [-6, 1, 0, 0, 0, -1, 0, 0, -2, 2, 0, 2], [2, 0, 0, 0, 0, -2, 0, 0, -2, 2, 0, 0], [3, -1, 0, 0, 0, -1, 0, 0, -2, 2, 0, -1], [0, 0, -1, 0, 1, 0, 5, 1, -1, -1, -2, 0], [0, 0, 0, 0, -2, 0, -4, 1, -1, -1, -2, 0], [0, 0, 0, 0, 0, 0, 5, -2, 2, 2, 0, 0], [0, 0, -3, 0, -4, 0, -3, 5, -3, -3, 1, 0], [-2, -1, 0, 2, 0, 1, 0, 0, -3, 3, 0, 0], [0, 0, 0, 0, -2, 0, -5, 1, -4, -4, -1, 0], [6, -1, 0, 1, 0, -4, 0, 0, 2, -2, 0, -1], [0, 0, -1, 0, -1, 0, 0, -4, 0, 0, 1, 0], [0, 0, -1, 0, 3, 0, 7, -1, 3, 3, 0, 0], [-11, 1, 0, -1, 0, 4, 0, 0, 0, 0, 0, 0], [-1, -2, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 3, 0, -1, -1, 0, 0, 1, 0], [0, 0, -5, 0, 1, 0, 4, -5, 1, 1, 1, 0], [0, 0, 4, 0, -2, 0, -1, 4, 0, 0, -2, 0], [-14, 0, 0, -4, 0, 0, 0, 0, 2, -2, 0, -2], [0, 0, -2, 0, 2, 0, 1, -4, 2, 2, 2, 0], [0, 0, 4, 0, 0, 0, 2, -4, -1, -1, 0, 0], [0, 0, -2, 0, -3, 0, -1, 4, 0, 0, 2, 0], [0, 0, -2, 0, -2, 0, 3, -3, 2, 2, 0, 0], [0, 0, 3, 0, 4, 0, -2, -1, 0, 0, -1, 0], [-13, -2, 0, -1, 0, -3, 0, 0, -1, 1, 0, 0], [4, 2, 0, 2, 0, 2, 0, 0, -2, 2, 0, 0], [10, -1, 0, -2, 0, 3, 0, 0, 1, -1, 0, 0], [-6, -1, 0, 2, 0, 3, 0, 0, -2, 2, 0, -2], [4, 0, 0, -1, 0, 4, 0, 0, -2, 2, 0, 1], [0, 0, 5, 0, 1, 0, 3, 1, -1, -1, 0, 0], [0, 0, 4, 0, -4, 0, -2, 5, -2, -2, 0, 0], [-7, 0, 0, 1, 0, -1, 0, 0, 3, -3, 0, -2], [0, 0, -1, 0, -5, 0, -6, -1, -1, -1, -1, 0], [10, -1, 0, -1, 0, 4, 0, 0, 3, -3, 0, -1], [-7, 1, 0, 3, 0, -2, 0, 0, -1, 1, 0, 2], [0, 0, -2, 0, -2, 0, -2, 5, -1, -1, 0, 0], [-8, -2, 0, 0, 0, 0, 0, 0, 4, -4, 0, 0], [0, -2, 0, -3, 0, 1, 0, 0, 2, -2, 0, -1], [-4, 0, 0, -2, 0, -1, 0, 0, 2, -2, 0, -2], [0, 0, -6, 0, -2, 0, -2, 0, 1, 1, 2, 0], [0, 0, -6, 0, 2, 0, 4, 1, 0, 0, 0, 0], [0, 0, -1, 0, -6, 0, 0, -1, -2, -2, 3, 0], [0, 0, -2, 0, 0, 0, -3, 2, -2, -2, -1, 0], [0, 0, 2, 0, 1, 0, -1, 2, -2, -2, -2, 0], [14, 1, 0, 3, 0, 1, 0, 0, -1, 1, 0, 1], [8, -2, 0, 1, 0, -5, 0, 0, 0, 0, 0, 1], [0, 0, 1, 0, 4, 0, -1, -3, -1, -1, 1, 0], [0, 0, -6, 0, 1, 0, 2, -4, 0, 0, 0, 0], [0, 0, 4, 0, -1, 0, -5, 8, -3, -3, 2, 0], [-12, 1, 0, -4, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, -4, 0, 2, 0, -4, 3, -2, -2, -2, 0], [1, 0, 0, -3, 0, 3, 0, 0, 2, -2, 0, -2], [1, 0, 0, -2, 0, -3, 0, 0, -2, 2, 0, 1], [0, 0, 2, 0, -4, 0, 3, 0, 2, 2, 2, 0], [0, 0, 1, 0, -4, 0, 0, 0, -4, -4, 2, 0], [-8, 2, 0, 1, 0, -1, 0, 0, 0, 0, 0, -1], [0, 0, 4, 0, -2, 0, 6, -5, 0, 0, 0, 0], [0, 1, 0, -2, 0, 1, 0, 0, -4, 4, 0, 2], [5, 0, 0, -5, 0, 0, 0, 0, 1, -1, 0, 0], [-1, 0, 0, 2, 0, 0, 0, 0, -1, 1, 0, 5], [1, 0, 0, 3, 0, 1, 0, 0, 2, -2, 0, 0], [0, 2, 0, 0, 0, 6, 0, 0, 0, 0, 0, 2], [0, 0, 2, 0, -1, 0, 3, 1, -2, -2, -2, 0], [-13, 2, 0, -1, 0, 5, 0, 0, -3, 3, 0, 0], [0, 0, 0, 0, 1, 0, 3, -4, 5, 5, 0, 0], [24, -1, 0, 2, 0, -3, 0, 0, 1, -1, 0, 0], [0, 0, 0, 2, 0, -2, 0, 0, -5, 5, 0, 2], [-12, 1, 0, 0, 0, -5, 0, 0, 1, -1, 0, 2], [0, 0, -5, 0, 3, 0, -3, 3, -2, -2, 1, 0], [-8, 2, 0, 3, 0, 4, 0, 0, -2, 2, 0, 5], [0, 0, 1, 0, -2, 0, -3, 6, 0, 0, 1, 0], [27, -2, 0, 1, 0, 3, 0, 0, -4, 4, 0, -2], [7, 0, 0, -2, 0, 0, 0, 0, -1, 1, 0, 1], [0, 0, -2, 0, -1, 0, -3, -2, -5, -5, 0, 0], [0, 0, -2, 0, 0, 0, 2, 0, 1, 1, 4, 0], [-4, -2, 0, -4, 0, -4, 0, 0, -2, 2, 0, 0], [0, 0, -6, 0, -2, 0, 1, -1, 2, 2, 1, 0], [10, 2, 0, 2, 0, -4, 0, 0, 0, 0, 0, 2], [0, 0, 0, 0, 7, 0, -4, 1, -2, -2, 0, 0], [-12, -3, 0, -4, 0, -7, 0, 0, 2, -2, 0, 0], [7, 0, 0, -3, 0, -1, 0, 0, -3, 3, 0, -2], [0, 1, 0, -2, 0, -1, 0, 0, -1, 1, 0, 2], [0, 0, 2, 0, 0, 0, 2, -5, 0, 0, 2, 0], [-1, 2, 0, -1, 0, 1, 0, 0, 2, -2, 0, 2], [-4, -1, 0, 0, 0, -3, 0, 0, 5, -5, 0, -2], [0, 0, 0, 0, -5, 0, 6, -3, 1, 1, 3, 0], [0, 0, -1, 0, 5, 0, -2, -1, -1, -1, 1, 0], [0, 0, 4, 0, -3, 0, 9, 3, -4, -4, -2, 0], [0, 0, -7, 0, 2, 0, -2, -4, -2, -2, 2, 0], [10, 1, 0, -3, 0, 0, 0, 0, -3, 3, 0, 3], [0, 0, 2, 0, -5, 0, -3, 6, -5, -5, -2, 0], [2, 2, 0, -7, 0, -2, 0, 0, 2, -2, 0, 1], [-6, -3, 0, -3, 0, 2, 0, 0, -2, 2, 0, -1], [8, 0, 0, -1, 0, 3, 0, 0, 4, -4, 0, -3], [0, 0, -6, 0, -2, 0, 0, -5, -1, -1, 0, 0], [6, -1, 0, 0, 0, -3, 0, 0, 2, -2, 0, 0], [-9, 0, 0, -2, 0, -4, 0, 0, 4, -4, 0, 3], [0, 0, -5, 0, 4, 0, 3, -8, 2, 2, 1, 0], [0, 0, -2, 0, 1, 0, -10, 2, 1, 1, -3, 0], [14, 0, 0, -4, 0, 0, 0, 0, 2, -2, 0, 0], [-14, -1, 0, -1, 0, 2, 0, 0, 1, -1, 0, 1], [9, -3, 0, -1, 0, -6, 0, 0, 5, -5, 0, 0], [0, 0, 2, 0, 2, 0, 2, 1, 0, 0, 0, 0], [2, 1, 0, 4, 0, 3, 0, 0, 3, -3, 0, 0], [0, 0, 0, -2, 0, 4, 0, 0, 2, -2, 0, -2], [16, 0, 0, -4, 0, -2, 0, 0, -3, 3, 0, -2], [0, 0, -5, 0, -3, 0, -5, 5, -2, -2, 1, 0], [-4, 2, 0, -1, 0, -3, 0, 0, 2, -2, 0, 3], [0, 0, -6, 0, 2, 0, -1, 1, -2, -2, -3, 0], [0, 0, -6, 0, -1, 0, -5, 0, -5, -5, 2, 0], [0, 0, -2, 0, 3, 0, -5, 2, 2, 2, 0, 0], [13, 0, 0, -3, 0, 5, 0, 0, 2, -2, 0, -2], [0, 0, -4, 0, 4, 0, 7, -11, 4, 4, 1, 0], [-1, -1, 0, -3, 0, 4, 0, 0, -2, 2, 0, -2], [0, 0, 0, 0, 2, 0, -4, 5, -1, -1, 0, 0], [0, 0, -2, 0, 3, 0, 2, -10, 1, 1, 1, 0], [0, 0, -2, 0, 7, 0, -2, -6, 2, 2, 2, 0], [0, 0, -1, 0, -3, 0, 2, 0, -2, -2, 1, 0], [-8, 4, 0, 2, 0, 4, 0, 0, -2, 2, 0, -2], [0, 0, -6, 0, -3, 0, -2, 9, -1, -1, 1, 0], [0, 0, 3, 0, -5, 0, -2, 5, -3, -3, 3, 0], [0, 0, 4, 0, 1, 0, 1, -5, -4, -4, -2, 0], [-1, -4, 0, 0, 0, -5, 0, 0, 2, -2, 0, 1], [0, -4, 0, -3, 0, -1, 0, 0, 0, 0, 0, -1], [-14, 0, 0, -4, 0, -1, 0, 0, -2, 2, 0, 2], [0, 0, 0, 0, -3, 0, -11, -1, -3, -3, 0, 0], [0, 0, 12, 0, -1, 0, 2, -3, 1, 1, 1, 0], [-26, 3, 0, 3, 0, 0, 0, 0, 3, -3, 0, 5], [0, 0, 0, 0, -2, 0, -11, -1, 0, 0, -1, 0], [0, 0, -6, 0, 2, 0, -8, 1, 1, 1, 0, 0], [0, 0, 6, 0, -1, 0, -2, 2, -1, -1, 3, 0], [-16, -3, 0, -7, 0, -1, 0, 0, 3, -3, 0, -3], [0, 0, 10, 0, -2, 0, 4, 6, -1, -1, -4, 0], [-12, -2, 0, -3, 0, -4, 0, 0, -2, 2, 0, -1], [-6, 1, 0, 0, 0, 1, 0, 0, 1, -1, 0, -4], [-8, 0, 0, 2, 0, 0, 0, 0, -3, 3, 0, -2], [15, 0, 0, -1, 0, 3, 0, 0, 3, -3, 0, 2], [-1, -3, 0, -1, 0, 0, 0, 0, 3, -3, 0, 2], [0, 0, -1, 0, 1, 0, -3, 1, -4, -4, -1, 0], [26, 0, 0, -3, 0, 2, 0, 0, 4, -4, 0, 1], [-6, -1, 0, 6, 0, 3, 0, 0, -1, 1, 0, -4], [0, 0, 8, 0, -6, 0, 4, 5, -5, -5, 0, 0], [0, 0, -1, 0, 7, 0, -1, -3, 2, 2, -3, 0], [0, 0, -3, 0, 8, 0, 6, 1, 4, 4, 1, 0], [12, 2, 0, 1, 0, 4, 0, 0, 0, 0, 0, 3], [6, 1, 0, 5, 0, -1, 0, 0, -3, 3, 0, 3], [0, 0, 1, 0, 2, 0, 3, -12, 0, 0, 1, 0], [-1, 0, 0, 3, 0, -2, 0, 0, 3, -3, 0, 2], [0, 0, 5, 0, -2, 0, 0, 1, 2, 2, 1, 0], [0, 0, -1, 0, 3, 0, -5, 3, -4, -4, 3, 0], [-12, 4, 0, 5, 0, -2, 0, 0, 0, 0, 0, -1], [-30, 0, 0, 2, 0, 6, 0, 0, -3, 3, 0, -2], [0, 0, -12, 0, 1, 0, -9, 0, 1, 1, 2, 0], [4, -3, 0, 0, 0, 1, 0, 0, -1, 1, 0, 0], [8, -1, 0, -2, 0, 1, 0, 0, 5, -5, 0, -2], [0, 0, -2, 0, 5, 0, 2, -6, 3, 3, 1, 0], [11, 2, 0, 0, 0, 6, 0, 0, -1, 1, 0, 3], [29, -1, 0, -2, 0, -4, 0, 0, 3, -3, 0, 3], [10, 3, 0, -3, 0, 2, 0, 0, 0, 0, 0, -5], [0, 0, 5, 0, -2, 0, -8, -1, 0, 0, 1, 0], [0, 0, 2, 0, -9, 0, 0, -1, 2, 2, 0, 0], [0, 0, -2, 0, 5, 0, 1, -2, 0, 0, 2, 0], [1, -2, 0, 2, 0, 3, 0, 0, -3, 3, 0, 1], [-4, -3, 0, -4, 0, -1, 0, 0, 0, 0, 0, -2], [0, 0, 6, 0, -3, 0, 0, 0, -1, -1, 1, 0], [10, -3, 0, -6, 0, -3, 0, 0, 5, -5, 0, 0], [-14, 4, 0, -2, 0, -4, 0, 0, 4, -4, 0, 4], [0, 0, 2, 0, -2, 0, -4, 4, -1, -1, 2, 0], [0, 0, -5, 0, 4, 0, 2, 0, 0, 0, -4, 0], [0, 0, 1, 0, -2, 0, 5, 7, -1, -1, 1, 0], [0, 0, 0, 0, 1, 0, 1, -10, 1, 1, -4, 0], [0, 0, -6, 0, -3, 0, 0, -4, 1, 1, 1, 0], [0, 0, 5, 0, -3, 0, -1, -1, -2, -2, -1, 0], [15, 0, 0, 2, 0, -4, 0, 0, -4, 4, 0, -1], [19, 1, 0, 6, 0, 8, 0, 0, -1, 1, 0, 1], [0, 0, -4, 0, 3, 0, -1, 4, -2, -2, -2, 0], [1, 3, 0, 6, 0, -1, 0, 0, -2, 2, 0, 3], [2, 2, 0, 0, 0, -2, 0, 0, -6, 6, 0, 0], [0, 0, -4, 0, 11, 0, -1, -2, 0, 0, 2, 0], [0, 0, 0, 0, -2, 0, 4, 0, -4, -4, 0, 0], [-8, 1, 0, -1, 0, 3, 0, 0, -1, 1, 0, -1], [0, 0, -4, 0, 7, 0, 12, -10, 7, 7, -1, 0], [9, -2, 0, 0, 0, -3, 0, 0, 8, -8, 0, 3], [-10, -5, 0, -6, 0, -5, 0, 0, 1, -1, 0, 0], [0, 0, -8, 0, 3, 0, -1, 6, -1, -1, -2, 0], [13, -2, 0, -5, 0, -5, 0, 0, -1, 1, 0, -4], [1, -3, 0, -1, 0, -6, 0, 0, 0, 0, 0, 2], [0, 2, 0, 2, 0, -4, 0, 0, -2, 2, 0, 6], [-10, -4, 0, 4, 0, -4, 0, 0, -1, 1, 0, -4], [0, 0, 2, 0, -6, 0, -1, -7, 2, 2, -1, 0], [18, -4, 0, -2, 0, 1, 0, 0, 0, 0, 0, -4], [-29, 3, 0, 6, 0, -2, 0, 0, 0, 0, 0, 1], [0, 0, -8, 0, -1, 0, -4, -3, 3, 3, -3, 0], [1, -1, 0, -3, 0, 2, 0, 0, 1, -1, 0, -2], [-4, 5, 0, -2, 0, 3, 0, 0, 3, -3, 0, 2], [-8, 3, 0, -1, 0, -3, 0, 0, 7, -7, 0, -1], [0, 0, 5, 0, -2, 0, -9, 6, -2, -2, 5, 0], [-17, 0, 0, -3, 0, -5, 0, 0, -3, 3, 0, -2], [26, 3, 0, 0, 0, 1, 0, 0, 3, -3, 0, 0], [0, 0, 6, 0, 2, 0, 10, -10, 5, 5, 4, 0], [0, 0, 6, 0, -7, 0, 2, 10, -2, -2, -2, 0], [0, 0, -1, 0, -6, 0, 6, -6, -2, -2, 6, 0], [0, 0, 1, 0, -3, 0, 4, 3, -1, -1, -1, 0], [0, 0, 1, 0, 2, 0, 5, -2, 4, 4, 3, 0], [0, 0, 6, 0, -2, 0, 2, -6, 6, 6, 0, 0], [0, 0, 4, 0, 4, 0, -3, -6, 2, 2, 0, 0], [0, 0, 0, 2, 0, -2, 0, 0, -4, 4, 0, -2], [0, 0, 6, 0, -10, 0, 0, 5, -4, -4, 0, 0], [0, 0, 8, 0, -5, 0, 3, 7, -5, -5, 4, 0], [0, 0, -9, 0, 0, 0, -5, 1, -3, -3, -3, 0], [0, 0, 4, 0, 7, 0, -2, -1, 6, 6, -4, 0], [0, 0, 2, 0, 6, 0, 13, -2, 6, 6, -2, 0], [0, 4, 0, 4, 0, 4, 0, 0, -2, 2, 0, -2], [0, 0, -4, 0, -1, 0, -5, -10, -1, -1, 2, 0], [0, 0, -10, 0, -2, 0, -8, 5, -4, -4, -4, 0], [0, 0, -3, 0, -3, 0, -3, 13, 1, 1, 0, 0], [8, 2, 0, -6, 0, -8, 0, 0, 0, 0, 0, 0], [0, 0, 5, 0, 1, 0, 6, -10, 2, 2, 3, 0], [0, 0, 10, 0, -1, 0, -1, -7, 2, 2, 2, 0], [13, 0, 0, 8, 0, 6, 0, 0, -2, 2, 0, -1], [0, 0, 10, 0, -1, 0, -6, 0, 0, 0, 0, 0], [-1, 0, 0, 3, 0, -3, 0, 0, 4, -4, 0, -2], [2, 1, 0, 3, 0, 4, 0, 0, -2, 2, 0, -3], [-18, 1, 0, -5, 0, -7, 0, 0, -1, 1, 0, 1], [-9, 2, 0, 10, 0, 8, 0, 0, 0, 0, 0, 3], [0, 0, 2, 0, 1, 0, 12, -7, 4, 4, 0, 0], [3, -4, 0, -5, 0, -5, 0, 0, 1, -1, 0, 2], [0, 0, -2, 0, -3, 0, -11, -9, 2, 2, -2, 0], [0, 0, -8, 0, -10, 0, -2, 13, -4, -4, 0, 0], [-26, 0, 0, -3, 0, -4, 0, 0, -4, 4, 0, 1], [0, 0, -14, 0, 2, 0, 6, -1, 2, 2, -2, 0], [0, 0, 14, 0, -3, 0, -2, 6, -2, -2, -2, 0], [10, -4, 0, -2, 0, 8, 0, 0, 2, -2, 0, -4], [0, 0, -11, 0, 0, 0, 5, -7, 1, 1, -1, 0], [0, 0, 4, 0, 2, 0, 3, -4, 4, 4, 0, 0], [9, -6, 0, -6, 0, 2, 0, 0, 5, -5, 0, -5], [0, 0, -2, 0, -7, 0, 9, 2, 0, 0, 0, 0], [2, 0, 0, 12, 0, 4, 0, 0, -2, 2, 0, 0], [0, 0, 1, 0, -5, 0, -1, 1, 3, 3, -2, 0], [12, -3, 0, 0, 0, -5, 0, 0, -1, 1, 0, 0], [0, 0, 8, 0, -5, 0, -3, -1, -7, -7, -4, 0], [0, 0, 12, 0, -4, 0, 4, 5, -5, -5, 2, 0], [0, 0, 2, 0, 3, 0, -2, 10, -1, -1, 3, 0], [18, 3, 0, -2, 0, -1, 0, 0, 1, -1, 0, 4], [-7, 2, 0, 9, 0, -5, 0, 0, -5, 5, 0, 2], [1, -4, 0, 7, 0, 1, 0, 0, -4, 4, 0, -4], [-27, 1, 0, 6, 0, -2, 0, 0, -4, 4, 0, 3], [16, 1, 0, 8, 0, 1, 0, 0, 7, -7, 0, -2], [0, 0, -6, 0, 7, 0, -4, -7, 1, 1, 3, 0], [11, -5, 0, -1, 0, -2, 0, 0, 0, 0, 0, -4], [8, -2, 0, 4, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, -16, 0, 0, 0, 0, -8, 4, 4, 2, 0], [24, -2, 0, -2, 0, -4, 0, 0, 4, -4, 0, -2], [-7, -1, 0, 3, 0, -2, 0, 0, -2, 2, 0, 2], [0, 0, 0, 5, 0, 10, 0, 0, 2, -2, 0, 3], [22, 3, 0, 1, 0, 5, 0, 0, 3, -3, 0, -1], [4, 2, 0, 14, 0, 2, 0, 0, -1, 1, 0, 4], [0, 0, 0, 0, 3, 0, -9, 0, 1, 1, 0, 0], [0, 0, 8, 0, 0, 0, 10, -5, 2, 2, 2, 0], [0, 0, 1, 0, -1, 0, -2, 1, -5, -5, 1, 0], [0, 0, 10, 0, -6, 0, -10, -2, -1, -1, 0, 0], [0, 0, -1, 0, 7, 0, -6, -10, 4, 4, -3, 0], [0, 0, 9, 0, 2, 0, 3, 6, 2, 2, 1, 0], [0, 0, -6, 0, 3, 0, 1, 4, 5, 5, -2, 0], [-19, 0, 0, 4, 0, -5, 0, 0, -2, 2, 0, -1], [0, 0, 5, 0, 0, 0, -8, 10, 2, 2, -2, 0], [0, 0, 12, 0, -2, 0, 9, -7, -2, -2, 1, 0], [-6, 0, 0, -9, 0, -1, 0, 0, 3, -3, 0, -1], [0, 0, -2, 0, 0, 0, 11, -5, 8, 8, 0, 0], [-47, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2], [0, 0, -20, 0, 1, 0, 5, -8, 3, 3, 2, 0], [2, 0, 0, 1, 0, 3, 0, 0, 2, -2, 0, -5], [-19, -6, 0, 2, 0, 1, 0, 0, 5, -5, 0, -3], [1, 1, 0, 7, 0, 4, 0, 0, -4, 4, 0, -2], [17, -6, 0, -3, 0, -13, 0, 0, 4, -4, 0, -4], [8, -2, 0, -8, 0, 0, 0, 0, -6, 6, 0, 2], [19, 1, 0, 2, 0, 8, 0, 0, 0, 0, 0, -3], [0, 0, 8, 0, 8, 0, -1, -13, 2, 2, 1, 0], [3, 5, 0, 5, 0, -6, 0, 0, 1, -1, 0, 4], [8, -1, 0, 6, 0, 3, 0, 0, -5, 5, 0, 2], [0, 0, -5, 0, -2, 0, 11, -8, 2, 2, 1, 0], [0, 0, 14, 0, 1, 0, 9, -3, 4, 4, -2, 0], [0, 0, 8, 0, -5, 0, 2, -6, 7, 7, 1, 0], [0, 0, -10, 0, -6, 0, -3, 0, 2, 2, -2, 0], [6, 0, 0, 4, 0, 13, 0, 0, -2, 2, 0, 2], [4, -2, 0, 4, 0, 4, 0, 0, -8, 8, 0, 0], [-11, -3, 0, 2, 0, -1, 0, 0, 0, 0, 0, -1], [-41, -1, 0, -1, 0, 0, 0, 0, -3, 3, 0, 2], [26, 2, 0, 2, 0, -2, 0, 0, 2, -2, 0, 0], [-37, 1, 0, 2, 0, 1, 0, 0, 1, -1, 0, -1], [0, 0, -6, 0, 5, 0, -2, -9, 8, 8, 2, 0], [0, 0, 2, 0, -3, 0, 8, 10, 0, 0, -2, 0], [-24, -3, 0, -2, 0, 3, 0, 0, -2, 2, 0, 0], [18, -2, 0, 0, 0, 6, 0, 0, 2, -2, 0, -6], [0, 0, 0, 0, 0, 0, -2, -8, 2, 2, 0, 0], [0, 0, 0, 0, 7, 0, -8, 2, 3, 3, -1, 0], [32, 2, 0, 2, 0, 0, 0, 0, -8, 8, 0, 2], [0, 0, 6, 0, 5, 0, 13, 1, 5, 5, 0, 0], [0, 0, 8, 0, 6, 0, 1, 11, 0, 0, -4, 0], [0, 0, 6, 0, -2, 0, 0, 6, -5, -5, 4, 0], [20, 1, 0, 5, 0, 4, 0, 0, -1, 1, 0, -1], [-1, -5, 0, -2, 0, 7, 0, 0, -3, 3, 0, -1], [0, 0, -8, 0, -3, 0, 0, 2, -3, -3, -3, 0], [26, 1, 0, 4, 0, 1, 0, 0, -7, 7, 0, 0], [1, 1, 0, -3, 0, -8, 0, 0, 3, -3, 0, 6], [0, 0, -18, 0, 11, 0, 8, -10, 4, 4, 0, 0], [0, 0, -6, 0, 10, 0, 3, -19, 6, 6, -2, 0], [34, -4, 0, -2, 0, -3, 0, 0, -4, 4, 0, -4], [0, 0, -5, 0, 4, 0, 12, -10, 4, 4, 2, 0], [-12, -2, 0, -7, 0, 3, 0, 0, -9, 9, 0, -1], [-3, -3, 0, 2, 0, 6, 0, 0, 0, 0, 0, -1], [-34, 0, 0, -1, 0, -4, 0, 0, 6, -6, 0, -9], [0, 0, 0, 0, -4, 0, 10, -6, 3, 3, 2, 0], [23, -3, 0, 2, 0, 5, 0, 0, 2, -2, 0, -7], [0, 0, 12, 0, 0, 0, -3, 6, 0, 0, 1, 0], [-6, -2, 0, 3, 0, -2, 0, 0, 10, -10, 0, -1], [0, 0, -6, 0, 4, 0, -8, 2, 4, 4, 0, 0], [0, 0, -2, 0, 1, 0, -3, -11, 0, 0, 2, 0], [0, 0, -12, 0, 2, 0, 9, -10, 6, 6, 2, 0], [0, 0, 8, 0, -8, 0, 9, -14, 0, 0, 1, 0], [40, 2, 0, 6, 0, 2, 0, 0, -2, 2, 0, 2], [0, 0, -6, 0, -5, 0, -8, -8, 2, 2, 0, 0], [17, 2, 0, -5, 0, 0, 0, 0, 2, -2, 0, 0], [0, 0, -1, 0, -8, 0, -8, 11, -10, -10, 5, 0], [-11, -5, 0, -2, 0, -2, 0, 0, -4, 4, 0, -1], [-10, -2, 0, -4, 0, 4, 0, 0, 4, -4, 0, -8], [0, 0, 9, 0, -10, 0, -9, 1, -1, -1, -1, 0], [-14, 2, 0, 4, 0, -7, 0, 0, -4, 4, 0, 2], [-27, -1, 0, 5, 0, 2, 0, 0, 0, 0, 0, 4], [0, 0, 4, 0, -1, 0, 5, 0, -1, -1, 0, 0], [0, 0, 4, 0, -2, 0, 3, -9, -2, -2, 0, 0], [0, 0, 0, 0, 0, 0, 4, -13, 4, 4, 2, 0], [-8, 0, 0, 3, 0, -5, 0, 0, -6, 6, 0, 1], [0, 0, -5, 0, 3, 0, -7, 13, -5, -5, 0, 0], [4, 2, 0, 2, 0, -6, 0, 0, -4, 4, 0, 6], [0, 0, 16, 0, 0, 0, 1, -2, -2, -2, 0, 0], [0, 0, 4, 0, 6, 0, -7, 1, -2, -2, -3, 0], [-4, 1, 0, -13, 0, -5, 0, 0, 1, -1, 0, -1], [16, -2, 0, -6, 0, -6, 0, 0, 0, 0, 0, -6], [0, 0, 0, 0, 4, 0, 2, 2, 1, 1, 2, 0], [-3, 3, 0, 11, 0, 6, 0, 0, 0, 0, 0, 4], [19, 1, 0, 5, 0, -2, 0, 0, -7, 7, 0, 4], [-13, 6, 0, 5, 0, 5, 0, 0, -6, 6, 0, 6], [-8, -1, 0, -4, 0, 3, 0, 0, -5, 5, 0, 4], [-40, 0, 0, -3, 0, 3, 0, 0, 1, -1, 0, -5], [30, -1, 0, 2, 0, 5, 0, 0, -8, 8, 0, 4], [0, 0, 2, 0, -5, 0, -12, -1, -5, -5, -7, 0], [0, 0, 0, 0, 4, 0, 3, -1, 6, 6, -1, 0], [0, 0, -6, 0, 3, 0, -9, 0, -4, -4, 4, 0], [-3, 4, 0, -6, 0, -2, 0, 0, 4, -4, 0, -3], [0, 0, -4, 0, -6, 0, -1, 1, 4, 4, -5, 0], [0, 0, 2, 0, 1, 0, 3, 3, 0, 0, 0, 0], [-12, 1, 0, 4, 0, -1, 0, 0, 8, -8, 0, 4], [-14, 4, 0, 0, 0, 0, 0, 0, 12, -12, 0, 0], [0, 0, 2, 0, 2, 0, -18, 6, -1, -1, 0, 0], [0, 0, -8, 0, -3, 0, 4, 12, -5, -5, -3, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4560_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4560_2_d_l();".
// 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_4560_2_d_l(:prec:=12)
    chi := MakeCharacter_4560_d();
    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_4560_2_d_l();".
// 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_4560_2_d_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4560_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2304, 0, 42864, 0, 40228, 0, 10480, 0, 1148, 0, 56, 0, 1]>,<31,R![-3392, 2168, 724, -256, -52, 6, 1]>],Snew);
    return Vf;
end function;