
// Make newform 2277.2.a.r in Magma, downloaded from the LMFDB on 20 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2277_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_2277_2_a_r();".
// 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_2277_2_a_r();".
// 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 := [1, -14, -35, 10, 36, -2, -11, 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, 1, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0], [2, 10, 3, -7, -1, 1, 0, 0], [-1, -2, 10, 4, -7, -1, 1, 0], [-2, -7, 7, 10, -6, -2, 1, 0], [3, 12, 10, -14, -11, 7, 2, -1], [2, -9, -16, 13, 12, -7, -2, 1]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 1, 1];
    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_2277_a();"
function MakeCharacter_2277_a()
    N := 2277;
    order := 1;
    char_gens := [254, 1036, 1684];
    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_2277_a_Hecke( : Kf := Kf)
    return MakeCharacter_2277_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]];
    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], [1, 1, 0, 0, 0, 0, 0, 1], [-1, 0, 0, 0, 0, -1, 1, -1], [-1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -1, 0, 1, 1, -1, 0], [0, 1, -1, -1, 1, -1, -1, -2], [-2, -1, -1, -1, -1, 1, -1, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 1, -2, 1, 1, 1, -2], [-3, 1, -2, -1, -1, 0, -1, 1], [-1, 1, 2, 4, -1, 2, -1, 4], [-1, -2, 2, 3, -1, 1, -2, -1], [-3, -4, 2, 2, 0, 1, 1, -1], [0, -1, 1, 4, -2, -1, 0, 0], [2, -2, 0, 0, -1, 2, 3, -1], [0, 1, 3, 0, -2, -1, 2, 2], [-2, 2, 2, 2, 0, -2, -4, 2], [-7, 1, 0, 1, -2, 0, 0, 0], [-4, 0, -2, 0, 0, 2, -2, 0], [-2, -2, 0, -2, 2, 0, 0, -4], [0, 1, 0, -4, -1, -1, 6, -1], [-1, -1, 0, 0, 1, 0, 3, 0], [1, 3, -2, 2, -2, 2, -2, 5], [-3, 1, 0, 0, -1, 0, 3, -2], [-5, 2, -2, -1, 1, -5, 2, -1], [-5, -1, -2, -1, -2, 0, -2, 0], [3, -1, 0, -2, 1, 0, 1, 0], [0, -4, 3, 1, -2, 4, 1, 3], [4, 0, 2, 3, -1, 0, -3, 2], [-1, 0, 3, -2, 3, -3, 1, -4], [0, 0, -3, -1, 4, 2, 1, -1], [2, -2, -4, -6, 1, -2, 1, -1], [-1, 4, -1, -2, 5, -3, -5, 2], [-2, 3, -1, 3, 1, -1, -3, 6], [-3, -4, 5, 2, -1, -1, -1, 0], [2, -4, -4, -2, 4, 0, 0, -2], [-7, 5, 0, 3, -1, 2, -1, 3], [0, -6, 1, -3, -2, -2, 3, -3], [-3, -2, 0, -3, -1, -5, 2, -5], [6, -1, 1, -4, 0, 1, 6, -2], [0, -8, -2, -6, 2, 2, 0, -4], [-4, 2, -2, -1, 6, -4, -4, -1], [-3, 0, -3, 4, -1, 3, -1, 2], [-5, 2, -2, 1, -5, -1, 0, 7], [2, 0, 2, -5, 3, 0, 7, 0], [-3, 6, -1, 4, -1, -1, -1, -2], [-9, 3, -2, -1, 3, -4, -1, -5], [2, -4, 2, -4, 4, -2, -2, 0], [1, -1, 2, 4, -3, 6, 3, 2], [-7, 2, -3, 6, 1, 1, -3, 4], [-8, -2, 6, 0, 2, -4, 8, -4], [0, 0, 3, -5, 4, -4, -1, -5], [6, -2, 4, 1, -2, 6, -4, 3], [4, -6, -2, -2, 0, 2, 6, 0], [4, -4, 4, 6, 2, -4, 0, -6], [-6, 0, -2, -2, -4, 4, -4, 4], [-3, -2, 3, 0, -7, -3, 7, 2], [-2, 6, -2, -4, 0, -6, 2, -2], [-2, -5, -3, -7, 5, -1, 3, -8], [-2, 5, 2, -2, 7, -3, 2, -5], [-3, -2, 1, 5, -6, 3, -4, 2], [-13, 2, -3, 2, 1, -3, 3, -2], [8, -1, 1, -6, 2, -7, 4, -6], [3, 5, -4, -2, -1, -2, -1, 8], [10, -6, 4, 0, 8, 2, -2, -4], [-5, -3, 6, 5, -3, 4, 5, 1], [1, -9, -1, -3, 3, -6, 6, -3], [-5, 4, -2, -3, 1, 1, 4, 11], [3, 6, -9, -4, -3, -1, -3, 4], [0, 4, 2, 0, 2, -4, 0, 6], [1, 3, 6, 4, -3, -2, 1, 4], [-7, 3, 0, 5, 4, -2, -2, -4], [5, -1, 2, 6, 3, -4, -1, 0], [-4, -6, 2, 7, 1, -4, -3, 0], [7, 9, -10, -3, 5, 2, -9, -3], [2, -2, 2, -1, 1, 8, -5, 6], [-3, 1, -4, 7, -5, 4, -9, 5], [-1, 1, 4, 1, -8, -4, 6, 2], [-3, -4, -1, 6, -5, 11, -7, 0], [4, 6, -2, 9, 2, -4, -4, 1], [10, 0, 0, -6, -2, 8, 0, 2], [5, 1, 0, 8, 3, 0, 1, 4], [-7, -5, 0, -4, -3, 2, 3, 0], [-1, -8, -1, -4, -1, -1, -3, -6], [16, 4, -2, -6, 4, -2, -4, -4], [0, 10, 4, 2, 0, 2, 2, 2], [1, -7, 9, 7, -3, 0, -4, 1], [9, 6, -5, -6, 1, -5, -7, 2], [3, -1, 0, -5, -1, 2, -1, 1], [-5, 1, 2, 7, -1, -2, -7, 1], [-9, 1, -6, -4, -3, 6, -7, 2], [-7, -3, 2, 1, 1, 0, 5, -7], [-7, 2, -6, -11, 3, 5, -2, 3], [3, -1, -2, -1, -3, 4, -11, 5], [-5, 9, -4, -2, -3, -2, 3, -4], [4, 0, -10, 0, -2, 8, -6, 0], [11, 5, 6, 2, -2, 6, -2, 7], [-14, 1, 0, 6, 1, 1, -6, -3], [-3, -4, -7, -6, -1, -1, -5, -4], [-9, 4, -3, 6, -3, 7, -3, 18], [-9, 10, 1, 3, 0, -3, 2, 2], [-3, 1, -12, -6, -3, 4, 1, -2], [-7, 2, -3, -11, 2, -5, 2, -12], [7, -6, -8, -6, 8, -3, -1, -3], [-11, -4, -3, 5, -1, -1, 3, 3], [-4, 6, -2, 0, 4, -2, 0, 4], [1, 4, 3, 2, -7, 3, 1, 4], [10, -10, -2, 4, -2, 10, 2, 2], [9, -6, -5, -4, 7, -3, 3, -10], [-5, 6, 7, 10, -9, 1, 1, 6], [4, -6, 1, -9, 8, -10, 3, -9], [-1, -5, 4, 2, -8, -4, 4, -3], [-6, 4, 0, -3, 3, 0, 3, -8], [0, 12, -2, 1, 5, -2, -3, 0], [8, -10, -2, -2, -7, 12, 1, 5], [-6, -4, 2, -7, -3, 0, 3, -2], [6, 3, -13, -12, 4, -5, 0, -6], [-10, 14, 0, -6, 6, -6, -6, 2], [11, -5, 6, 12, -5, 4, 3, 2], [2, 6, 4, 14, -4, -2, -4, 0], [3, 2, 7, 4, 1, -5, 9, -12], [0, -5, 3, 15, -11, 9, -1, 12], [-2, 2, 6, 2, 6, 2, 4, -10], [-9, 13, -6, -5, 1, 8, -3, 7], [-11, -6, 1, 11, 0, 7, -10, -2], [7, -5, 4, -10, 11, -4, 7, -6], [-4, 3, 5, 6, -6, 3, -4, -2], [-11, 5, -6, 3, -2, 4, -4, 4], [9, -9, 3, 1, 7, -2, -2, 1], [-5, 0, -9, -4, 3, -5, -1, 0], [1, 3, 6, 8, -7, 10, -5, 14], [0, -18, 2, 1, -7, 4, 7, 4], [-6, 12, -2, 4, 2, 4, -6, -4], [-7, 6, 2, 3, 3, -9, 4, 9], [5, 9, -9, -19, 9, -12, 2, -9], [13, 3, -12, -5, 4, 4, -10, -6], [-11, 4, -5, -11, 0, 7, -4, 6], [7, -3, -8, -17, 6, -8, 12, -10], [5, 0, -4, -1, -3, -9, 8, -15], [-11, 4, 7, 4, -1, 1, 5, 14], [1, -6, -2, -11, 3, -3, 2, -7], [-4, 12, -2, 6, 2, 2, 2, 6], [-10, 6, -2, 0, -4, -4, -12, 4], [-9, -14, 3, 5, -3, 5, -3, -3], [7, 1, -6, 5, 3, 2, -15, 3], [4, 12, 6, 2, -4, -8, 4, 8], [3, -16, -2, -3, 1, 5, 4, -15], [-8, 6, 0, 8, -4, 0, 4, -2], [-4, -3, -1, 2, 2, 7, 0, 10], [0, 0, -2, 14, 0, 0, -10, 6], [12, 2, 2, 6, -3, -8, 5, -7], [-3, 15, -6, -11, 3, -12, 3, 1], [-13, -6, 7, 6, 1, 9, -1, -4], [1, -3, 4, 1, 4, 8, -2, 10], [-4, 10, -10, -3, -6, 12, -4, 13], [-13, 2, -4, -6, -4, 1, 3, -7], [-24, 0, 4, 4, -4, 0, 6, -2], [5, -1, 1, -7, 5, 2, 10, 3], [-5, -8, 5, -1, -2, -1, 10, 0], [20, 1, 1, -10, 2, -7, 6, 6], [-7, 16, 1, -1, 2, -7, 2, -4], [13, -4, 9, -2, -7, 11, -7, 6], [19, 5, 6, 5, -3, 2, -1, 5], [-1, -3, 6, 8, 2, -2, 10, 5], [5, 1, 0, -7, 11, -4, 5, -3], [-1, -9, 0, -1, 1, 10, -3, -9], [1, 6, 1, -6, -9, 1, 9, 0], [-3, 7, -14, -12, 9, -8, 1, 0], [14, 14, -8, -2, -2, 0, 2, -2], [11, -3, -2, 2, -3, 6, 5, 12], [16, 18, -4, -4, -2, -12, -2, 4], [-5, -5, 18, 6, -3, 0, 15, 8], [-2, 6, 3, 9, 0, -8, -1, -5], [-4, -8, 4, 2, -2, 4, 6, -12], [8, -6, -10, -10, -1, 4, -5, -9], [3, 6, 13, 18, -5, -3, -3, 8], [-12, 0, -2, 8, -4, 2, -4, 14], [-11, -2, -7, -5, 6, 5, 2, 2], [13, -3, 7, 7, -1, 8, -8, 7], [-4, 2, 8, 2, -6, 4, -2, -2], [-5, -19, 4, -2, -11, 10, -3, -2], [9, -15, 8, -1, 3, -4, 3, -11], [5, 18, -3, -4, -5, 1, 3, 24], [25, 5, 0, 0, 1, 2, 5, 4], [0, -2, -2, -2, 3, -8, -9, -5], [-2, 4, -10, -10, 12, -4, -4, -4], [-32, -6, -4, -7, 5, -12, -5, -14], [2, -2, 4, 4, -12, -6, 2, 4], [25, -12, 0, -11, 11, -1, -6, -13], [-3, -1, 8, -13, -1, -8, 27, -9], [-20, 10, 0, -2, 8, -10, 2, -2], [-8, 6, 4, -9, -7, 0, -3, 2], [-8, 12, 8, 10, -4, 8, -8, 10], [16, -4, 4, 6, 10, -4, -8, 2], [6, 6, -2, -9, 17, -8, -5, -14], [1, -11, 0, 0, -3, 6, -7, -16], [20, -4, 10, -2, -4, 14, 4, 4], [-12, -21, 9, 15, 1, 5, -7, 0], [-4, -2, 7, -1, 0, 0, -3, -9], [-19, 2, -11, 1, -4, 7, -10, 20], [5, 4, 1, -6, 11, -3, 5, 2], [-9, 5, 4, 1, -3, -16, -7, -3], [-9, 21, -6, 4, 3, 0, 3, 4], [-13, 5, -10, 8, -9, 16, -13, 4], [6, -22, 4, 0, -8, 16, -4, -6], [11, 5, 0, -7, 10, -10, 10, -14], [1, 2, 5, -10, -5, 3, 1, 0], [23, 10, 4, 5, -11, 11, 0, 13], [9, -1, 8, -3, 11, -18, 7, -9], [2, 10, -10, 0, 10, 0, -4, -4], [-8, -2, 14, -2, -4, -2, 2, 0], [3, -13, 8, -3, -2, -16, 8, -10], [1, 7, 2, 9, 9, 6, -13, 7], [-4, 8, -5, 1, 12, -2, -13, 1], [13, -5, 14, 6, 1, -2, -1, 0], [-11, -2, 1, 6, -7, 11, 1, 20], [-9, -2, 2, 5, -1, -9, 4, -5], [10, -12, 2, 15, -2, 12, 4, 9], [-12, -6, -8, -10, -4, -8, 0, -24], [5, 0, 1, 4, -1, 7, 1, -12], [4, -3, -8, -10, 1, -5, 8, 7], [12, -6, 10, -6, -6, -12, 8, -10], [0, -8, -4, -7, -11, -6, 13, -10], [-14, 8, -5, 11, 6, -4, -13, 7], [15, -16, 5, 10, 1, 1, 9, 6], [23, 9, -6, -6, 10, -2, -10, -9], [-10, -1, -3, -14, 4, -5, 8, 2], [3, 14, -9, -3, 2, 11, -6, -2], [-6, 0, 2, 16, -12, 8, -6, 8], [14, 19, -11, 2, 12, -1, -8, 6], [14, 11, -8, -2, -5, 7, -6, 15], [13, 2, 9, 7, -2, 9, -10, 6], [-13, -8, -1, 6, -5, 7, -1, 14], [-20, 12, -2, 3, -3, -10, 1, -6], [32, 0, 12, 6, -2, -4, 4, 2], [10, -8, 4, -2, -2, -4, 4, 6], [-8, -1, -5, 5, 1, 13, -11, 4], [12, 4, -12, 0, -8, 4, -4, 4], [-22, -5, 5, -4, 8, 7, 4, -4], [13, 19, -6, -11, -5, -2, 1, 13], [5, -8, 3, -8, 3, 1, 15, 8], [-8, -6, -8, 14, 2, 4, -18, 2], [5, 2, 11, 3, 11, 5, -1, -3], [-7, 2, 4, 15, 1, 3, -8, -1], [-18, 8, 3, 3, 2, 4, 3, 11], [-11, -3, -12, -15, 15, -2, -5, 1], [0, -4, -12, 5, 0, 14, -2, 3], [-4, 3, 6, -4, 7, -7, 14, -5], [-28, 6, -4, 10, -22, 6, -2, 10], [8, -12, -6, 6, 2, 2, 10, -14], [0, -7, 13, 9, -11, 3, 23, -6], [9, -11, 4, 6, -1, 6, -9, 8], [21, 1, -8, -8, 3, -8, -3, -8], [5, -4, 6, 11, -9, -9, 4, -13], [0, 4, 9, -1, -10, 0, 1, 15], [3, -7, 2, -8, 1, 6, 3, 4], [-33, 4, 1, 17, -10, -3, 6, 6], [10, 8, -8, -8, 4, -6, -4, 8], [-3, -8, -13, -4, -5, 11, 1, 6], [-9, 6, -13, 1, -10, 11, -14, 22], [-6, 14, -2, -12, -2, -4, 0, 8], [4, -6, -10, -4, 0, 10, 10, 2], [17, -13, 9, 1, -7, 10, 2, -13], [-8, -8, 12, 10, 0, -4, 14, -8], [4, 16, -16, -11, 15, -2, -5, 0], [15, 9, 0, 13, 0, 4, -18, 2], [-15, 3, 4, 13, 6, -8, 0, -10], [-12, 1, -12, 0, -13, 7, 6, 3], [-13, -2, 15, 8, 1, 3, 1, 12], [21, -15, -2, -9, 11, 2, 7, 3], [18, 0, -4, 1, 3, 8, -19, 6], [6, 6, 12, 8, 6, -10, -8, -2], [-8, -9, 8, -2, -1, -7, 0, -15], [-10, 18, -4, -2, -2, -20, 6, -10], [-20, 4, -12, -4, 0, 4, -8, 20], [6, -1, 7, -8, 12, -3, 8, -2], [16, 24, -2, -2, 0, -16, 4, -8], [-2, 12, 5, -5, -6, -2, -3, -5], [10, 26, 6, 10, -6, -4, -8, 10], [-17, 2, 8, 5, 1, -9, 12, -9], [1, 25, -2, 3, -2, -4, 2, 10], [-1, 3, -14, -14, -1, 0, -17, 8], [7, -15, -10, -8, -5, 4, 11, 0], [-13, -2, 13, -2, -5, 3, 7, -10], [6, 3, 13, 14, 0, -5, -10, 0], [-15, -9, -4, 10, -5, 10, -7, 12], [5, -24, 9, -11, -4, 9, 0, 8], [4, -12, -2, -4, -4, -6, 16, -14], [5, -12, 0, -13, 5, 15, 12, 1], [-9, -11, 0, -3, -8, 0, 6, -18], [-15, -3, 0, -1, -3, 8, -5, -7], [15, -4, 15, 6, -9, 7, 3, -2], [17, -9, 12, -8, 7, -14, 7, -16], [3, -14, 15, 4, 9, -13, -1, -18], [5, -14, -9, -21, 4, -21, 16, -22], [-6, -12, -10, -19, 3, -8, 5, -12], [-2, 10, -10, -8, -2, 4, 14, 6], [8, 0, -20, -12, 14, -2, -8, -16], [-17, 12, 7, 18, -7, 13, -17, 10], [0, -4, -10, -22, 14, -10, -6, -30], [21, 6, 1, -7, 12, -17, 2, 8], [44, 1, 10, 12, 9, -5, -6, -3], [-4, 9, -7, -10, 22, -9, -10, -18], [-7, -13, 6, 1, 4, -10, -12, -8], [-5, 2, 3, 13, 1, 1, -19, -7], [10, -14, -10, -10, -2, -14, 16, -26], [10, 4, -14, -4, 2, 8, 4, -14], [-3, -7, 8, 19, -9, 16, -3, 7], [7, -2, 11, -6, 5, 7, 7, -12], [-5, 20, -4, -9, 17, -11, 6, -3], [35, 1, -2, -8, 18, -2, 2, -11], [-9, -8, 7, 15, -8, -1, 4, 0], [-1, -17, 2, 11, 3, 16, -9, 11], [15, -3, -4, 2, 11, -6, 3, -8], [10, 22, -6, 2, 0, -2, -12, 14], [16, -6, 4, 6, 8, -6, 14, 6], [-15, -16, -12, -7, -1, 3, -10, 11], [18, -16, 2, -12, -4, 12, 0, 2], [-41, -4, -2, -11, 1, -15, 0, -23], [-3, -22, 8, 2, -4, 11, 13, 7], [-12, -4, 14, 10, -16, 2, 8, 8], [1, -8, 2, -1, -5, 15, 8, -21], [3, -17, 6, 15, -5, 16, 3, -5], [-5, 2, -11, -17, 2, -3, -12, -8], [10, 2, 10, 0, -8, 10, 6, 18], [6, 0, 0, -17, 5, -20, 19, -18], [-14, -10, 10, 0, 2, 10, -2, -10], [39, -5, 8, -2, -7, -4, 15, -10], [-12, 9, 19, 15, 3, -9, -7, 16], [24, -20, 0, 0, -12, 8, 16, 12], [-4, -14, 10, 6, 0, 0, 2, 8], [17, 6, -13, 12, -13, 5, -9, 6], [-2, -4, -17, -17, 4, -6, 15, -9], [12, 31, -11, -5, 9, -1, 1, 0], [-12, -9, -7, 8, -10, 1, 0, 28], [23, 7, -4, -10, 19, -8, 1, 0], [-4, 1, -3, -11, 7, 7, 5, 0], [34, -1, -5, -18, 8, -15, 4, -4], [-16, 5, -1, 3, 1, 9, -11, -12], [3, -12, 1, 9, -7, 7, -19, -15], [-9, -11, -4, 11, -10, 24, -16, 10], [-13, -2, 4, -7, 11, -11, 4, -7], [6, 12, 2, 9, -1, -20, 3, 2], [9, -17, 4, 3, 8, -12, 6, -34], [-8, -8, -3, 19, -2, 4, -11, 15], [-3, 10, -13, -1, 15, -11, -5, 5], [3, 5, -2, -13, -3, 4, 1, 15], [14, -16, 22, 20, -6, 6, -10, -4], [5, -11, 4, -14, -4, -20, 16, -25], [-15, -12, -7, -6, -1, 7, 7, -20], [5, -12, -3, -7, 3, -13, 19, -25], [-8, -26, 2, -4, -14, 4, 10, -6], [0, 17, 8, 2, 5, -5, -8, -9], [2, 0, 1, -7, -8, -22, 11, -21], [5, -16, 18, 9, -7, 3, 2, -5], [22, -14, 4, 5, -16, 26, -2, 17], [-16, -6, 4, -14, 8, -14, 20, -12], [-13, -4, 8, 17, -9, -3, 4, 27], [2, -7, 15, 0, 4, 7, 22, -14], [-10, -6, -1, 25, -12, 8, -21, 11], [-17, -2, -2, 13, -3, 17, -20, 19], [-10, -16, 16, 6, -2, 8, -4, -6], [-34, 8, -14, -6, -2, 2, -6, 18], [-17, -19, 12, -15, 15, -14, 15, -35], [19, 3, 10, 3, -11, 10, -15, 7], [5, -8, 10, 11, 9, 5, -20, -1], [0, -7, 11, 9, 3, 9, 3, -8], [-11, -19, 3, 19, -5, 4, -16, -13], [8, -1, -5, 16, -10, 17, -14, 4], [-11, -15, 16, 15, -9, 6, 7, 11], [10, -18, -9, 17, -8, 16, -3, 25], [-3, 6, -1, 13, -6, 15, -8, 20], [-11, 12, 0, 14, -10, 9, 5, 25], [31, -7, 14, -5, 1, 18, 5, 13], [20, -2, 2, -10, 4, 12, -14, 16], [29, 3, -4, -30, 13, -10, 5, -20], [-15, 1, 4, 6, -1, 2, -13, 16], [-39, -6, 3, -1, -2, -5, 16, -14], [29, 2, 15, 10, -13, -7, 7, -4], [22, 4, 18, 16, -1, -8, -1, -13], [12, 8, 0, 22, -7, 2, 5, 5], [-21, 24, 9, 2, -11, -1, 5, 6], [-17, -24, 1, -6, 7, 13, -5, 4], [-19, 8, -4, 22, 2, 13, -7, 17], [2, 8, 14, 10, -6, -2, 8, 12], [-23, 7, -14, -1, 3, -24, -1, -17], [-33, 3, 4, -1, 1, -20, 17, 5], [-37, 2, -17, -5, 4, -21, -8, -10], [-17, 19, -8, -10, 5, -6, -3, 12], [-7, 5, -22, -14, -9, 12, -5, 2], [14, -16, 12, -4, -20, -4, 14, -6], [-6, -17, 15, 2, -4, -11, 16, -8], [-31, -27, 16, 19, -1, -6, -9, 3], [-25, 1, 10, -11, 9, -14, -1, -7], [-15, 1, -3, 3, -11, 4, 4, 1], [9, -26, 14, 0, 0, 11, -9, -3], [-1, -5, -16, -2, -14, -12, -14, -1], [-3, 5, -18, 7, 6, 0, -26, 6], [1, -10, -16, 3, 1, -1, 0, -1], [-14, 10, 12, 28, -2, 4, 0, 26], [12, 2, -8, -2, -10, 20, -26, 18], [-18, 15, -10, -16, 15, -1, 0, 13], [22, 16, 20, 17, -9, -6, 13, -2], [-27, -6, -14, 1, 17, -5, -14, -11], [13, -10, -15, -28, 15, -15, 3, -24], [7, 1, -26, -12, 5, 2, -5, -20], [-1, 18, -9, -24, 13, -13, 5, -12], [-55, -9, -20, -21, 12, 14, 0, 0], [-14, 13, -5, -18, -4, 3, 2, -4], [7, 18, -1, -4, 1, 3, 17, 12], [-11, 8, 4, -31, 11, 3, 14, -11], [20, -22, 18, 7, 10, 20, 0, -5], [14, 13, 5, -5, 11, -5, 15, 8], [-18, 6, 11, -5, 2, 0, 17, -9], [12, -14, 6, -8, 18, 2, 2, -18], [27, -22, -1, 2, -11, 25, 3, 12], [26, 0, 2, -10, -8, 22, -4, 8], [25, 5, -8, -6, 8, 8, -20, -13], [-20, 0, -6, 4, -10, 8, -12, -2], [8, 0, -8, -22, 17, -14, -3, -19], [-39, 17, -3, -5, -3, -4, 12, 13], [-20, 6, -28, -2, 20, 0, -16, -4], [21, 20, 7, 10, -1, -11, -19, 12], [16, -8, 18, 4, -6, 14, 10, 20], [-18, -24, 10, 4, -6, 10, -2, 8], [-37, 15, 4, 8, -1, -14, 11, 4], [-1, 12, 1, 6, 11, 5, -1, 0], [-43, -15, 4, 1, -5, 2, 11, 1], [5, 0, 14, -1, -9, -9, 0, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2277_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2277_2_a_r();".
// 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_2277_2_a_r(:prec:=8)
    chi := MakeCharacter_2277_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_2277_2_a_r();".
// 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_2277_2_a_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2277_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 14, -35, -10, 36, 2, -11, 0, 1]>,<5,R![2, -26, 80, 8, -81, 26, 14, -8, 1]>,<17,R![544, 3296, -5552, 560, 1282, -58, -74, 0, 1]>],Snew);
    return Vf;
end function;