
// Make newform 5577.2.a.y in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5577_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_5577_2_a_y();".
// 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_5577_2_a_y();".
// 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 := [7, -7, -33, 13, 21, -7, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [-3, -1, 1, 0, 0, 0, 0], [2, -5, -1, 1, 0, 0, 0], [4, 3, -6, -1, 1, 0, 0], [-2, 9, 4, -7, -1, 1, 0], [-3, -14, 11, 12, -7, -2, 1]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 2];
    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_5577_a();"
function MakeCharacter_5577_a()
    N := 5577;
    order := 1;
    char_gens := [3719, 508, 1354];
    v := [1, 1, 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;

function MakeCharacter_5577_a_Hecke(Kf)
    return MakeCharacter_5577_a();
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], [1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, -1], [1, 0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [-1, 1, -1, 0, 0, 0, 1], [1, 1, 0, 0, 0, -1, 0], [2, -2, 0, -1, 1, 0, 0], [-2, 1, 0, 0, 0, -1, 1], [-1, -2, -2, 0, 0, 0, 1], [1, -1, 0, -1, -1, 1, 2], [1, -1, 1, -1, 0, 0, 2], [-3, 1, -1, 0, -2, 0, -1], [1, 1, -2, 1, -1, 1, 0], [2, -1, 1, 1, 1, 0, -2], [2, -4, -1, -1, 0, 1, 0], [-1, 0, 0, -1, 0, 0, 0], [1, 0, 3, 1, 0, 1, -1], [5, 1, 3, 0, -1, 0, 0], [5, -4, -2, -2, -1, 0, 0], [1, -1, 0, -1, -2, 1, 1], [4, 1, 0, 0, 1, -1, 2], [6, -1, 1, -2, -1, 0, 1], [3, -1, -1, 0, -1, -2, 0], [-1, -3, -1, 2, 0, 0, -1], [0, -3, -1, -3, 1, 0, 2], [0, -4, 1, -3, 0, -1, 2], [1, -4, -2, 0, -1, 0, 2], [2, 3, 1, -1, -1, 0, -2], [2, -1, 2, 0, -2, -1, 1], [-2, 2, 0, 2, 2, -2, 0], [0, 1, 2, -1, -1, 1, -1], [0, 1, 1, 1, -2, 2, 1], [5, -4, -1, -1, -1, 1, 0], [7, 1, 2, 0, 0, -3, 0], [4, 2, 4, -1, 3, 0, -2], [-6, 7, 3, 4, -1, -2, 1], [4, 2, 0, 2, -2, -2, -2], [1, 1, -4, -1, 0, -1, 3], [2, -1, 5, 2, 2, -2, 2], [4, 2, -2, -3, 1, -2, 0], [-2, 1, 3, -1, -3, 0, -2], [3, 2, 0, 4, -1, 0, 0], [3, -3, 1, 1, 2, 2, -2], [8, -1, 1, -3, -1, 0, -4], [-5, -3, -1, 2, 0, 4, -3], [-1, 0, -3, -5, 0, 3, 1], [-3, 2, -1, 1, 1, -3, 4], [1, 1, -1, 2, -1, 0, -2], [4, -7, -1, -1, -2, 0, 3], [3, 2, -3, -1, -3, 3, 2], [-8, -3, -1, -1, 3, 0, -2], [-2, -3, -3, -3, -3, 2, -2], [8, -7, 1, 1, -1, 2, 4], [-2, 2, 1, 3, 2, 1, 2], [2, 0, -2, 0, 2, -2, 0], [1, 4, 2, 4, -1, -2, 0], [-8, 2, 3, 3, -2, 3, 0], [3, -6, -1, -5, -1, 3, 4], [0, -3, -2, -4, 4, 1, -1], [-3, 3, 1, 3, -2, 0, 0], [-1, -5, -4, 0, 4, -3, 0], [-12, 2, 0, 5, -1, -4, 2], [2, 3, -5, 2, 2, -4, -2], [18, -1, 2, 3, 3, -1, -1], [4, 3, -4, 1, -3, 1, -1], [-2, 0, -5, 3, 0, 1, -2], [-7, -2, -2, 3, -2, 0, 2], [2, -1, 1, -5, -1, -4, 4], [-1, -10, -3, 3, 0, 1, -1], [-5, 6, 3, 3, 5, 1, -4], [2, -1, -3, 1, -3, 0, 4], [-2, -8, -2, 0, 4, -2, -2], [-10, 3, 3, 0, 1, -2, -5], [2, 0, -1, -1, 0, 5, -4], [-2, -2, 6, 4, -2, 0, -2], [-11, 9, 6, 1, -3, 1, -2], [9, -2, -1, -3, 2, -1, 3], [7, -2, 4, 2, -1, 0, 6], [4, 1, -1, 4, 4, -2, -4], [8, -2, 1, -3, -2, -3, 6], [0, 4, -8, -2, 0, -4, 2], [10, -4, -2, -2, 4, -2, 4], [-1, -7, -5, -4, 0, 0, -1], [-10, -2, -4, -4, -2, 2, 2], [9, -2, 4, 0, 2, 2, -1], [-16, 14, 5, 5, 2, -3, -2], [5, 3, -1, 1, 4, -2, -2], [6, -3, 6, 5, -1, -3, -5], [16, -4, 2, 1, 3, -4, -2], [9, 2, -3, -1, 3, -1, 6], [18, -1, -2, 1, -1, 3, -3], [-5, 4, -1, 4, 0, 1, -6], [-3, -8, 0, 0, -2, 4, 1], [4, -2, -3, 5, 2, 1, -6], [2, 5, 1, -6, -1, -2, -5], [8, 6, 0, -2, -2, 2, 4], [11, -5, 0, -1, -4, 1, -7], [-6, 10, -1, -1, 0, -1, 2], [-8, 5, 1, 5, 0, -2, -5], [3, -2, 1, -1, 3, -3, -2], [-10, 8, 5, -1, 2, -1, -4], [-5, -1, 0, 5, 2, 5, -5], [11, -9, -1, 0, 0, 6, -3], [-2, -4, -1, 5, 0, 1, 2], [4, -2, 4, -2, -2, -2, 2], [5, 1, 5, 3, 2, -2, -2], [6, 3, -5, 1, -1, 2, 0], [-16, 0, -5, 1, 2, 1, 4], [4, -9, 2, 4, 0, 3, 1], [-13, 0, 0, 2, -1, 2, -8], [9, 2, 5, -5, -4, 1, 1], [-16, 5, -4, -3, -7, 3, 1], [16, -1, 1, 2, 5, 0, -3], [0, 6, 2, 4, 4, 0, -2], [5, -2, 0, -8, 2, -2, 3], [2, -7, -1, -1, -3, 4, -6], [6, -3, -3, -1, -5, 2, 0], [5, -10, 1, -2, -6, 1, 4], [-8, 8, -1, -1, 2, 1, -8], [-9, -2, -1, -6, -2, -1, 6], [-18, 1, -1, 3, 0, -4, 3], [13, -13, -5, -4, 5, -2, 4], [-2, -11, 4, 1, 1, 3, -1], [-5, 9, 8, 1, 0, -5, -1], [-9, -5, -2, -5, -7, 3, 4], [14, 6, -2, 0, -2, -4, 0], [-8, 12, 0, 4, -2, -4, 6], [3, 6, 8, 2, -5, 0, -2], [-1, 4, 6, -2, 3, -2, -4], [22, 7, 0, -4, -1, 1, 0], [-2, -1, 3, -1, -7, 2, 0], [-12, 8, 6, -5, -3, -4, 0], [-3, -5, -13, -1, 2, -2, 6], [-16, -8, 7, 1, 0, 1, 0], [12, -5, 3, 2, 3, 2, -3], [21, 2, 2, -4, -3, 4, -4], [0, -7, -1, -3, -3, 0, 12], [15, -1, 0, 7, 4, 1, 3], [-1, 11, 6, 2, 2, -5, 2], [-15, 6, -1, 5, 3, -5, 0], [-6, 6, 6, 6, 0, 0, 4], [-17, 4, 1, -7, -5, 1, 4], [6, -2, -6, 6, -2, 0, 4], [14, 6, -8, 2, -6, 2, 4], [-2, 0, -7, -1, 0, -1, 2], [-4, -5, -6, 2, -8, -1, 9], [-2, 6, 4, -4, 0, 2, 0], [4, -2, 5, -1, 2, 1, 4], [4, -3, -7, 1, -3, -2, 8], [20, -10, -4, -4, 6, 2, 2], [-10, 7, -1, -5, -5, -2, -2], [-5, -6, 8, 1, -2, 6, -4], [10, 1, 1, 5, -1, 2, 4], [-10, 11, -3, 1, -1, -4, 14], [-18, -3, -5, -1, 0, 0, 9], [6, -1, -11, 2, 7, 0, 5], [5, 0, 6, 3, 2, -6, 6], [-3, -5, -9, 1, -2, 2, 4], [-5, 3, -7, -4, -9, 4, -2], [-8, 1, 9, -5, 2, 2, 9], [9, 3, 10, 4, 4, 3, -8], [10, 4, 2, -7, 1, 0, -2], [6, -7, 2, -3, -3, -3, -3], [6, 2, -4, -4, 4, -2, -8], [10, -2, -6, 0, 2, 0, 4], [8, -8, 3, -9, -2, -3, 6], [-5, -6, 4, -2, 5, 4, 0], [-26, 3, 1, -1, -3, 0, 4], [6, -1, 2, 2, 3, -1, -12], [-15, 5, -3, 2, -1, -2, 12], [7, 10, -6, 5, -4, 0, -12], [18, -7, 1, -9, 3, 2, -2], [6, -3, 5, 5, -3, 2, 2], [-22, -5, -9, -1, 3, 6, -4], [-15, -4, 6, 2, 1, 2, 8], [4, 1, 8, 3, 1, -5, -5], [-7, 5, 11, 2, 6, -4, -7], [-2, 10, 12, 6, 6, -2, -2], [-5, 10, 4, 12, 2, 2, -3], [26, 12, -3, 3, -2, 3, -4], [0, -2, 0, -1, -5, 0, 6], [-5, 1, 5, -3, -8, 6, 4], [3, 1, 10, 5, 1, 1, -4], [-14, 11, 1, -1, 1, 0, 8], [-19, 2, -3, -4, 4, 9, 2], [2, -13, 3, -2, 3, 2, -1], [0, -1, 5, 5, 3, 0, -12], [24, -9, -6, -2, -3, 7, 8], [4, 8, -6, -1, -3, 8, 0], [-14, -8, -2, 6, -2, 2, -4], [22, -11, 3, -3, -3, 0, 4], [4, -9, -1, 6, 9, 2, -1], [-6, 6, -2, 4, -2, 4, 4], [0, -3, -9, -7, 5, -4, -4], [8, 12, -1, -3, 0, 5, 0], [0, 11, -1, -1, -3, 2, -8], [21, 4, -3, 1, -3, -5, -2], [-5, 8, -1, 2, -4, 1, 4], [9, 4, 3, 9, 3, -3, -14], [-8, 7, -6, 6, 2, -1, -7], [-12, 4, 3, 1, 6, 1, 6], [-9, 4, 2, 3, 6, -2, 0], [-13, 7, 1, 6, 5, 4, -6], [-6, -6, -8, 4, -2, -2, 0], [-26, 13, 5, 4, -9, 2, -3], [-10, 14, -5, -1, -2, -1, 4], [-15, 6, -1, -7, -1, 3, 0], [14, -3, -9, -1, -3, 0, 2], [-5, -2, 9, 4, 6, -3, 10], [-6, 8, -6, 0, 8, 2, -2], [-2, 13, 1, 3, -1, 2, -2], [20, 0, 4, -4, -4, 4, -4], [-11, 18, 5, 3, -5, 3, 2], [-1, -19, 0, 5, 3, -1, -4], [-1, -17, 3, 2, 4, 6, -9], [-4, 13, -6, 0, -8, 1, 13], [9, 9, 2, -1, -1, -3, 6], [3, 4, 8, 4, -2, -4, -5], [10, -6, -4, -4, 2, -6, 8], [-25, 11, -2, 0, 0, -5, 2], [5, 10, 15, 9, 2, 3, -9], [-5, 20, -1, -1, -2, 1, 5], [-3, 2, -1, 0, -2, 7, 4], [14, 4, 11, 1, 2, 1, 2], [-24, -6, -8, 2, 2, -2, 4], [14, 5, 5, -3, -3, 2, -2], [21, -15, 0, -3, -2, 3, 5], [-22, 3, 0, 2, -3, 5, 8], [8, 10, 0, 2, -2, -2, 0], [-19, -9, 3, -6, 2, 2, -1], [-3, 13, -6, 2, -2, 1, -6], [-4, 7, 11, -7, -2, 2, 3], [16, -5, -1, -3, -1, -6, -6], [16, 0, 0, -4, -4, 0, 4], [-16, 20, 7, 11, -2, 5, -12], [-15, 5, 5, 5, -4, 6, 2], [-8, 5, 7, -2, -4, -10, -2], [20, -12, -4, -9, -1, -6, 10], [32, -6, 3, 3, 4, -5, 0], [-8, 9, 6, 8, 8, 5, -13], [8, -10, -2, -4, 0, 0, 8], [-30, -8, 7, 5, -2, 1, -4], [-25, 1, 5, 5, 2, -2, 2], [-11, -18, -6, -13, -8, 4, 8], [12, 3, -2, 3, 7, -1, -3], [5, -3, -2, -11, 3, -3, -8], [17, -14, 0, 6, 7, -2, 2], [-13, 14, -1, 0, 0, 7, -4], [6, -5, 3, -5, -9, -2, 2], [-7, -7, -15, -9, 2, 6, -2], [9, -19, 2, -1, 7, 1, -14], [6, -5, 5, -2, 10, -2, -12], [-4, 11, 3, -3, -14, -2, -1], [37, -8, 3, -1, 3, 7, -2], [-16, 17, 1, -8, -2, 4, 6], [-17, 11, 8, 8, 2, -7, 2], [14, -6, -10, -4, 4, 4, 0], [8, -3, -13, 1, 3, 4, 8], [14, -23, -13, -7, -8, 0, 15], [19, -8, -7, -2, -2, 3, -8], [-6, -9, -5, 3, 13, -6, 8], [-14, 8, 0, -6, -8, 4, 6], [-2, 21, 2, 3, 5, -3, -9], [-12, 16, -2, 0, -4, 6, 2], [17, 8, 1, -5, -12, 7, -5], [29, -12, -11, -1, 2, 7, 9], [3, 1, -2, -6, -6, -3, -4], [16, -1, -6, 2, -12, -5, 9], [8, -3, 1, 3, 7, -6, 2], [-15, 4, -1, 3, 4, -3, 5], [-12, 8, -8, 6, 0, 0, -10], [-12, -3, 9, -9, 4, 2, 1], [-12, -6, 13, 5, -8, 1, -10], [24, -1, 5, 0, -5, -8, 5], [9, 9, 11, 6, 13, -4, -16], [-23, 8, -4, 4, 1, 6, -10], [-35, -6, -2, 12, -5, -6, 4], [-17, 2, 18, -1, 2, 4, 0], [-21, 16, 13, 8, 2, -5, 6], [6, 17, -6, 5, -11, -3, 1], [-9, -4, 4, 3, 6, -4, -8], [-25, -4, -10, -6, -5, 4, 2], [-31, -11, -16, 0, 2, 3, -2], [-16, 24, 4, 0, 4, 4, -4], [0, -2, 2, 8, 10, 0, -4], [1, -10, -7, -5, -5, -7, 8], [32, 5, 1, -10, 3, -6, 13], [28, 5, -1, -3, 8, -8, -1], [-18, 15, 7, 11, 5, -6, -10], [-24, 12, 4, 5, 5, -2, -10], [15, -4, 3, 3, 3, -7, -10], [-5, -12, -15, -1, 1, 1, -8], [15, -2, 5, -2, -2, -3, -8], [29, -9, 5, 2, 6, 0, 7], [6, 4, 4, 8, 6, -4, -4], [18, -7, -13, -3, -9, 8, -4], [-3, 2, -1, 18, -2, -5, -10], [35, -5, 8, 2, 4, 5, -14], [-24, 10, 7, 7, 2, -1, 0], [3, 9, 7, 1, -6, -4, 8], [16, -23, -7, -5, 6, -2, -1], [-16, -2, -10, -1, 5, -6, 4], [-8, -17, -5, 1, 5, -6, 4], [-9, 5, 8, -7, -7, -5, 2], [-6, 5, 5, 10, 6, 0, -20], [8, -15, -15, -5, 3, 6, 2], [18, -18, 0, -4, -4, -2, -4], [2, 8, -11, 1, 0, -1, 14], [26, 12, -2, -6, 0, 10, -10], [-1, -2, -3, 3, 2, 9, 1], [-3, 1, 1, -14, -6, 2, 1], [16, 9, 3, 7, -7, -4, 8], [24, -13, 8, -4, -9, -3, 2], [-16, 10, -7, -9, -10, 1, 6], [-10, 12, -6, -4, 0, 2, -4], [-3, 0, -7, 7, -7, 1, -2], [-16, -4, 3, 5, 4, -3, -16], [8, 13, -3, 5, 6, -2, -5], [-12, -3, 9, 7, 2, 2, -5], [6, -2, -2, -4, 10, 4, 0], [-6, 9, 0, 12, -4, 7, -7], [4, -3, 5, 6, 3, -2, 13], [47, -14, -13, 1, -1, 7, 6], [22, 4, -6, 13, 5, 4, 0], [-27, -12, 0, 2, 4, -4, -3], [-16, 21, -3, -6, -11, -6, 11], [-16, 0, -14, -4, 4, -6, 2], [18, -17, 3, -11, -10, 6, -11], [9, 4, -4, 4, 7, -2, 12], [30, -14, 8, 0, -4, 6, 2], [6, -7, -20, -2, 2, 3, 11], [-1, -8, 5, -3, -9, 1, -12], [11, -7, -4, 8, 2, 7, -8], [10, 2, -5, 5, -8, -5, 0], [-13, -1, 1, 0, -3, 0, 8], [3, -8, -5, -11, -3, 5, -10], [-16, 4, -12, -10, -4, 8, -2], [8, 11, 8, -4, 0, -3, 7], [3, 3, 6, 11, 3, -9, -2], [14, -4, -4, -2, -8, -4, 14], [27, 4, 9, -5, -1, -1, 12], [-2, 20, -4, 2, 8, -8, 0], [6, 3, 3, 8, -7, -10, -1], [-6, 2, 9, 1, -8, -3, -18], [8, 6, 22, 6, 2, -8, 2], [12, -13, -8, -7, 5, -3, 11], [11, -20, -7, -11, -9, -3, 12], [-14, -2, 2, 8, 4, 4, -16], [-17, 8, -6, -4, 7, 6, 2], [20, -10, -9, -5, -4, -5, 0], [-27, 10, 5, 11, -8, -3, 5], [-3, 11, 1, 4, 10, 0, -9], [-31, 18, 9, 8, 2, 5, 0], [-15, 17, 0, -7, -10, -1, -13], [-7, 15, 4, 9, 11, -7, 0], [13, -4, 13, -7, 1, -7, -4], [-13, -7, -12, 7, -2, 3, 7], [39, -21, -9, -8, -3, 2, -6], [20, -6, 3, 11, 10, -5, -14], [-2, 9, 7, -3, 4, 0, -19], [20, -3, 3, -1, 5, 0, 12], [-25, 4, 1, -5, -7, -3, 4], [-2, 0, -6, -2, 4, -6, 14], [-28, 26, 8, 7, 1, 4, -4], [-35, 10, 11, 5, 5, -1, 2], [15, 14, 6, -2, -15, 4, 4], [15, 0, -4, -7, 8, -8, 2], [-30, -1, 7, 3, 7, -2, 2], [-28, 3, -3, -9, -5, 4, 10], [31, -14, -13, -5, -1, 5, -2], [-13, 1, 7, -8, -12, 0, -3], [-17, -17, 15, 4, -2, -2, -1], [-10, 12, 16, 10, -8, 0, -2], [-9, 19, 6, 9, 3, 3, -4], [34, -30, -8, -4, -2, -6, 6], [-1, 0, 8, -4, 11, -6, -10], [-12, -1, -23, -7, 4, 8, 7], [-7, 2, 23, 0, 4, -5, -8], [53, 0, 0, -6, 4, -8, 5], [-23, 4, 0, -18, -8, 0, 9], [15, 10, 6, 0, -1, 0, 4], [23, -13, 13, 3, -6, -8, -6], [31, -14, -2, 2, 7, -4, 6], [3, 15, 14, -3, -3, 3, -2], [1, -16, -19, 3, 6, -5, -3], [-14, -3, 1, 9, -5, 10, -2], [3, -22, -6, -6, 1, 4, 4], [12, -15, 11, -6, -2, 6, 4], [27, -1, 0, -5, 9, -3, 2], [-12, 27, -1, 12, 5, -6, -3], [-19, 13, 6, 7, 3, 5, -18], [12, 0, -12, -8, 0, 4, 4], [-2, -2, 4, -4, 14, 6, -6], [11, -18, -5, 2, -8, -5, 8], [15, -1, 10, -2, 4, 3, 0], [-30, -6, 12, -2, 6, 6, 2], [6, -3, -3, -15, 5, 2, 2], [5, -8, -7, 1, 1, 15, -2], [0, 20, 1, 9, 0, -5, -6], [2, 15, -7, 3, -11, 12, 0], [33, -18, 2, 8, 11, 4, 10], [34, -9, -14, -7, 5, -1, 11], [29, -1, 19, 3, 0, 2, -4], [10, 7, 2, -8, 0, -1, -5], [-13, -11, -9, -10, 1, 0, 10], [7, 20, 5, 7, 3, 3, -20], [19, -14, -7, -3, 4, 5, -11], [37, -13, -13, 6, -5, 2, 4], [-40, 6, 10, 8, -2, 0, -8], [-5, -9, -8, -13, 4, -7, 3], [-4, -5, 3, 1, -9, 6, 6], [-9, -5, 10, -5, 12, -5, -5], [40, 6, 4, -2, 0, -2, -6], [8, -15, -15, 7, -1, 6, -6], [19, -22, -11, 7, 1, 3, -2], [32, -18, -4, -10, 8, 2, -8], [-27, -20, 1, -1, 2, 7, -15], [19, -18, -4, -19, 4, 6, -2], [-14, 17, 1, -1, -11, 2, -18], [-28, 7, -3, 5, -5, 8, 8], [9, -11, 12, -9, -3, -5, 2], [2, 18, 9, 3, 2, 9, -14], [-2, -2, -23, -3, 0, 9, -2], [-11, 8, -5, 3, 0, 13, 1], [-12, 14, 16, -4, -10, 2, 0], [-41, 4, -1, 5, 1, 1, -6], [26, -21, -8, -5, -3, 5, 19], [-23, 0, -9, 2, 8, -3, -10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5577_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5577_2_a_y();".
// 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_5577_2_a_y(:prec:=7)
    chi := MakeCharacter_5577_a();
    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_5577_2_a_y();".
// 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_5577_2_a_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5577_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![7, -7, -33, 13, 21, -7, -3, 1]>,<5,R![-8, 144, -198, -32, 74, -6, -6, 1]>,<7,R![496, 160, -524, -16, 136, -18, -6, 1]>],Snew);
    return Vf;
end function;