
// Make newform 2277.2.a.m in Magma, downloaded from the LMFDB on 23 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_m();".
// 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_m();".
// 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 := [-2, -7, 6, 10, -4, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [1, 5, -2, -3, 1, 0], [-3, -1, 9, 0, -4, 1], [-1, -5, 7, 1, -4, 1], [-5, -3, 10, 0, -4, 1]];
    Rf_basisdens := [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 := [[-1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, -1, 0, 0, -1, 1], [0, 1, -1, 0, 1, 0], [-1, 0, 0, 0, 0, 0], [1, -2, 1, 2, 0, 0], [0, -1, 2, 2, 1, -1], [-1, -1, -1, -1, -2, -1], [1, 0, 0, 0, 0, 0], [-2, -1, 2, 1, 0, -3], [-2, 1, 1, -3, 0, 2], [-1, 0, -1, -3, -1, 0], [-2, 1, -1, -3, 0, -2], [1, 2, 1, 3, -1, 0], [-5, 5, 0, -2, 1, 0], [2, 0, -1, 2, 2, -1], [-7, 1, 1, -1, 0, 1], [-1, -2, -1, 3, 3, -2], [0, 4, -4, 0, 2, -2], [-5, 1, 1, -4, -3, 3], [-4, 2, 0, -3, -1, 0], [-1, -1, -4, 1, 4, -4], [-7, -1, 1, 1, 2, -3], [1, -3, 0, 1, 2, 2], [6, -5, 5, 2, -3, 4], [1, -4, 0, -1, -3, -1], [6, 4, -3, -2, 2, 3], [-4, 3, -2, -2, 3, -1], [-8, -1, 3, 6, -1, -2], [4, 0, 1, 0, -2, 5], [3, -4, 0, 2, -2, -3], [-6, 1, -1, -1, 0, 2], [-5, 2, -1, 1, -1, 4], [0, -3, 0, 3, 2, -1], [2, 0, -1, 2, 0, 1], [-2, 4, 4, -5, 1, 0], [3, -3, -1, 1, -4, 5], [7, 1, 5, 0, 1, 7], [-11, -1, 3, 3, 2, 1], [4, -7, 3, 4, -1, 0], [1, 5, -4, 2, 1, -2], [-1, 1, -6, -1, 0, -2], [1, 8, -5, -1, 5, 0], [-9, 4, 0, 0, 2, -3], [1, 6, 1, 2, 0, -4], [-6, -3, -2, 2, -3, -5], [-5, 8, -3, -7, -3, 6], [-4, 9, 1, -6, 5, -4], [-7, -5, -2, 3, -4, 2], [-6, 1, -4, -6, 3, 1], [8, -1, -1, -5, 4, 2], [2, 0, -8, -3, 7, -4], [-6, -5, 0, 2, -1, 1], [0, -1, 0, -2, -7, 7], [-13, -2, 0, 0, 2, -7], [-4, 1, -3, 0, -1, -4], [-1, 4, 2, -2, -6, 5], [-8, 4, 5, 2, -4, 3], [-1, -1, 2, 8, -3, -2], [10, -4, -4, -8, -2, 10], [-15, 8, -2, -7, -1, 7], [1, 8, -2, -3, 1, -1], [12, 3, 6, 0, -1, -3], [7, 2, 3, 8, 10, -6], [8, 4, -9, 2, 6, -3], [6, -4, -5, 2, 0, 1], [5, 6, 0, -4, 4, 7], [-11, 7, -9, -9, 2, -5], [3, -3, 6, 7, -6, 2], [10, -12, 2, 7, -9, 6], [8, -5, -3, 1, 4, -2], [-11, 7, 8, -1, 0, -4], [6, -9, 1, 2, -5, 8], [-11, -1, -1, -9, -6, -1], [5, 0, -8, -3, -1, 1], [-9, -4, -6, -5, 5, -3], [-2, -10, -4, -2, -10, 4], [1, -1, -9, 2, 3, -3], [-3, -4, -10, -3, 1, -1], [2, -2, -6, -5, -3, 4], [-4, 0, 1, 10, -2, -11], [-4, 2, -8, -2, -6, 10], [-7, -2, -1, -5, -5, -2], [3, -11, 10, 3, -2, -4], [11, 0, -1, 0, -2, 2], [-10, 0, 2, 9, 1, -8], [8, -3, 0, 2, 3, -9], [1, -3, 1, 9, 6, -5], [-7, -3, 2, 4, 3, -6], [-3, 5, -4, 1, 12, 2], [-10, -3, -4, 0, -9, 5], [20, -2, -1, 8, 8, -9], [0, 7, -2, 3, 6, -7], [0, 4, 5, -1, 3, 5], [-7, -1, -3, 4, 7, -1], [2, 0, -5, -10, -6, 13], [17, -7, 0, 4, 9, -6], [19, -17, 4, 11, -10, 6], [-2, -6, -2, 4, -2, -2], [3, 6, 12, -4, 4, -9], [-10, -5, 4, -7, -2, 3], [-19, 8, -11, -9, 1, -4], [-19, 0, 4, -3, -1, 5], [6, -7, 4, 6, -5, 9], [15, 10, -6, 9, 7, 1], [-10, 12, -7, -11, -3, 5], [0, -11, 2, -1, 4, 3], [6, -7, 7, 6, -11, 2], [-6, 16, 1, -4, 10, -7], [-6, 8, -6, -9, 7, 2], [24, -14, 9, 6, 2, 1], [11, -10, -1, -1, 3, 6], [14, -4, 2, 8, 2, 0], [-5, 10, 6, 11, 1, -7], [-3, -13, 10, 9, -4, 4], [4, -6, -1, 2, -6, -3], [0, -2, 9, 10, 8, 3], [10, -14, 11, 3, -5, 5], [0, 7, 5, -1, -2, -2], [-8, -9, 1, -4, 1, -2], [-16, 11, -1, -12, -5, 4], [16, -14, -1, 9, 5, -7], [11, -1, 6, 1, 0, 8], [-13, -1, 3, 10, -3, 3], [-9, -1, -10, -13, 0, 0], [-5, -1, 4, -3, 0, 10], [-4, -1, 5, 0, -3, -8], [12, 1, -9, 10, 5, -14], [-5, -13, 2, 3, -16, 4], [-7, 2, 4, 3, 9, 3], [-1, 3, -8, -14, -11, 4], [-1, -4, 2, -3, 7, 5], [-3, -3, 15, -1, -4, 3], [6, 5, -1, -4, -1, 6], [-10, 1, 10, -7, -2, -1], [-10, 6, 13, 8, -4, -7], [-21, 19, -6, -9, 0, 10], [-12, -11, 3, -2, -17, 4], [-14, -5, -3, 10, 1, -12], [15, -7, -1, 15, 0, -5], [-7, -5, -5, -4, -11, 7], [-4, 2, 1, -12, -8, 7], [-14, -1, 2, 3, -10, 3], [-14, 20, -11, -4, 4, 7], [-9, 14, 5, -1, 9, 2], [-6, 7, -6, -8, -9, -1], [-4, 4, 17, 8, 6, -3], [4, -4, 8, 1, -7, 10], [5, 2, 3, 0, 2, -8], [11, -4, 3, 6, -6, 0], [-18, -7, 7, -2, -15, 6], [-19, 21, -1, -11, 4, -1], [15, -10, 15, 1, 3, -6], [6, -7, 5, -1, -12, 4], [-15, 3, 0, 3, -6, 6], [-6, 7, -3, -6, 1, -6], [-14, -14, -3, -14, -2, 3], [14, 4, -9, -3, 9, 3], [7, -5, 0, -5, 10, -2], [-7, 11, 8, -1, 4, 6], [-22, 4, -5, -9, -11, 7], [0, 7, 10, 6, 1, 3], [-19, 17, -3, -6, 13, 3], [7, 5, -9, -13, 4, 9], [9, -18, 16, 5, -3, 5], [-15, 12, 3, -19, -1, 12], [-5, -6, -3, 5, 3, -12], [12, -10, 11, 8, -8, 9], [-16, -2, 8, -2, 2, -6], [-5, -4, -6, 4, -12, 5], [-11, 3, 0, 9, 2, -16], [14, -15, -6, 3, -10, -7], [19, -13, 8, 1, -8, 0], [6, -8, -1, 8, 0, -5], [-21, 3, 0, 6, 7, -2], [13, 1, -3, 5, 12, -11], [-3, 2, -10, -16, -6, -5], [-12, 14, 9, -1, 9, -7], [-23, 20, -4, -21, 1, 1], [11, -10, 6, -11, 1, 5], [-6, -8, 24, 1, -15, -2], [-13, -2, 0, 11, -7, -5], [-13, 19, -4, -20, 7, -2], [-19, -5, 2, -1, -4, 0], [-11, 3, 9, 7, -6, -5], [3, -7, -2, -11, -12, 12], [6, 7, -16, 4, 3, 7], [23, -9, 1, 7, 4, 3], [6, -4, 13, 9, -1, 11], [4, 8, -14, -4, -4, 22], [5, -10, -13, -6, 6, -4], [-5, 5, 8, 3, 6, 12], [-11, 0, -7, -1, 7, -12], [-2, 3, 4, -20, -9, 19], [-7, 7, -6, -11, 4, -8], [-2, 3, -5, -12, 13, 0], [-12, -8, 2, 12, 2, 2], [25, 0, 14, 9, -5, 13], [14, 5, 3, 8, 3, 10], [2, -4, -11, -19, 1, -1], [-15, 15, 10, 13, 8, -22], [1, 18, -3, -9, 13, -10], [-1, 13, -9, -14, -3, 1], [-7, 20, -14, -11, 9, -5], [-2, -11, 2, 2, -5, 1], [0, 13, 5, -3, 14, 8], [0, 7, -4, 0, -9, 3], [2, -4, -13, -5, 11, 5], [-3, 5, -7, 8, 9, -17], [-14, -3, -3, -1, 10, -10], [-5, 5, -3, 8, 1, 1], [41, -13, 3, 12, 5, -3], [5, -21, 14, 5, 0, -8], [-19, -2, -3, 13, 3, -18], [-7, -1, 15, 23, 2, -3], [-6, -2, -13, 8, 14, 1], [-10, -6, -10, 7, 11, -16], [-19, -7, 26, 4, -13, 2], [-12, 6, 6, -16, -6, 12], [-1, -14, 2, 8, -6, 11], [24, -16, -15, -11, -9, 13], [15, 13, -9, 1, 0, 15], [8, 8, -8, -9, 5, -8], [6, -1, -15, -2, -1, 0], [-23, -10, -5, -6, 2, -2], [7, -8, -4, -6, 0, 1], [-16, 5, -16, -12, 11, -7], [8, 2, -24, -15, 3, -6], [-2, 1, 13, 2, 3, -8], [-28, 22, 8, 4, 2, -2], [19, -4, 22, -3, -5, -1], [12, 10, 2, -10, 6, 6], [-30, -6, 17, 20, -12, -5], [7, -9, -13, -5, 2, -3], [12, -5, 0, 24, -1, -1], [-24, 1, -2, -6, -1, 3], [12, -19, 13, 2, 3, -4], [1, -9, 0, 5, -6, -10], [51, -8, -4, -2, -4, 3], [11, -14, 18, 8, -8, -9], [4, -8, 26, 0, -4, 6], [-31, -7, 9, -12, -13, 3], [14, 4, -3, 13, 11, 1], [32, 0, -3, 21, 1, 7], [17, 7, -3, -15, -8, 13], [8, 19, -1, 9, -4, 10], [2, -12, 10, -1, -1, 16], [-1, -14, 13, 23, -11, -4], [0, 7, -19, -4, 9, 10], [19, -6, 13, 5, 1, 22], [5, -6, 6, -9, 7, 7], [22, -20, -1, 20, -4, 7], [-4, -18, 21, 8, -10, -9], [-7, 3, -9, -23, -4, 7], [-22, -13, 1, 10, -1, -6], [23, -3, 12, 8, 1, 6], [19, -10, -7, 9, -9, -2], [7, -10, 6, 3, -3, 1], [0, -6, 4, 23, -5, -10], [-5, -2, 10, 8, 8, 1], [-30, 17, -1, -4, -9, 12], [-19, 20, 11, -9, -1, -2], [17, -20, 9, 25, -3, -2], [-21, 31, 2, -3, 10, -14], [-11, -3, -6, -21, -18, 22], [-4, -14, 3, 4, -8, 11], [-11, -2, 4, 9, -1, -5], [3, -18, -3, -3, -25, 6], [-4, -19, -2, -14, -3, 5], [30, -20, -7, 15, -3, -9], [3, 12, -6, 12, 0, -3], [-17, -7, 3, 11, -12, -3], [16, -23, 7, 14, -9, -6], [36, 4, 0, -14, 2, 4], [-22, -15, 2, 7, 4, 7], [31, -28, 5, 6, -10, 10], [-4, -16, -2, -16, -8, 14], [22, 9, -4, 4, 21, -11], [8, 11, -18, -12, 9, -7], [-8, 4, -2, -24, -18, 12], [-13, -1, -7, -2, -21, 11], [1, -2, -9, 3, 5, -4], [5, 7, 2, -3, 4, -8], [-21, 11, 5, 3, 6, -1], [-42, 13, 1, -4, 3, -6], [2, 6, 0, -6, -4, -16], [47, -23, 5, 23, 6, -9], [5, -13, 12, 15, -4, -2], [3, 10, 4, -5, -17, 5], [9, 4, 6, -21, -3, 13], [6, -2, -15, 6, -2, 17], [-22, 11, 8, -12, -17, 13], [-23, 21, -17, -19, -2, 1], [6, -9, 27, 2, -9, 10], [7, -15, 8, -12, -9, 10], [7, 27, -18, -7, 16, 12], [13, -12, 6, -5, -17, 9], [-6, 12, -20, -17, -3, -10], [5, -14, 18, 14, 8, -9], [42, 3, 10, 1, -10, 5], [-8, -5, 4, -18, -11, 21], [14, -3, 10, 0, 11, -19], [-20, 17, 11, -8, -13, 6], [-25, -6, -6, 4, 4, -25], [-38, 29, -4, -14, 3, -1], [-25, 1, 19, 24, -9, -11], [-1, 20, 4, -1, 1, -5], [22, -5, 17, 0, 1, -2], [16, 9, 4, -4, 3, 15], [24, -2, -12, -2, -8, -4], [23, -14, -15, -12, 4, 2], [6, 12, -13, -18, 4, -7], [4, 12, -18, 14, 12, -22], [31, -1, -15, 4, 23, -11], [5, 7, 6, 15, -2, -12], [2, -11, -21, 2, 7, -8], [-4, -9, 10, 9, 0, 9], [21, -13, 16, 16, 19, -22], [-1, 7, -12, -9, 0, 0], [50, -6, 5, 5, 1, 1], [3, -3, 5, 17, 20, -15], [33, -1, 5, 5, -2, 23], [-12, 16, 14, 5, 9, -24], [-19, -11, -7, 11, 4, 1], [-28, -1, -1, -6, -9, 18], [-18, 14, 6, 10, -12, -4], [4, 5, 3, -10, -9, 4], [15, -28, 14, 18, -4, -3], [7, 0, -8, 11, -9, 9], [19, -3, 11, 15, 8, -17], [1, -14, 20, 14, -2, -7], [21, 18, -15, -28, 4, 14], [-27, -3, 1, -18, 1, -1], [-36, 14, 9, -5, 1, -9], [-4, 11, 4, -22, 7, -11], [-14, 10, 6, 7, -3, 6], [6, -19, 11, 28, 11, -18], [36, 11, -22, -6, 9, -5], [37, -29, 17, 14, -5, -5], [-19, 14, -10, -11, 3, -13], [-12, -4, -3, -3, 13, -1], [7, 11, -12, 7, 10, 2], [2, 24, -7, -18, 4, -5], [-20, 2, 19, 12, -10, 1], [50, -10, 10, -6, -4, 20], [35, -23, 6, 7, -6, 2], [14, 11, -12, -3, 2, -3], [-52, 34, 5, -18, 6, 1], [-16, -1, -28, 6, 11, -11], [21, 21, -10, -11, 10, 4], [-6, 16, 6, -5, -7, -14], [22, 13, -4, -9, -6, -9], [4, -21, -6, 2, -7, -1], [31, 10, -13, 7, 13, -2], [-4, -1, 13, 32, -17, -2], [-33, 5, 22, 13, 0, -8], [-19, 29, 19, 13, 8, 1], [2, 20, -12, -18, 12, 16], [24, -6, 9, 21, 9, -17], [5, 30, 7, 7, 13, -12], [-1, 10, 3, 5, -3, 10], [21, -19, 5, 38, -9, 3], [20, 10, -16, 9, 7, -22], [-18, 13, -8, -6, 27, -1], [-12, 30, -13, -23, -7, 7], [-22, 11, -4, -1, 4, -3], [-15, -4, -13, -4, -12, 8], [20, -8, 1, 10, 6, 17], [-17, 20, 2, -11, 7, -17], [-24, 40, -12, -11, 17, -12], [11, -9, 4, 27, 20, -10], [23, -15, 10, -6, -5, 8], [20, -15, 8, 10, 9, -23], [25, 9, 2, 21, 6, 14], [-4, 22, -24, -9, 23, -2], [-2, -15, -1, -7, 0, 8], [-5, 7, 9, -11, 2, 3], [-21, 22, -5, -25, 1, 0], [-4, 3, 0, -28, 7, 5], [-13, -27, 22, 6, -27, 4], [-15, -12, 13, 15, -15, -8], [-8, 23, 8, 2, 27, -5], [23, 16, -15, 1, 5, -14], [1, 17, 1, -12, -15, 7], [-20, 20, 3, -8, -4, -9], [6, -23, 8, 15, -24, 3], [7, -11, -11, 8, -3, 7], [5, -5, 26, -5, -4, -4], [-16, -3, -18, -12, 15, -15], [-45, 18, 5, -5, 5, 8], [22, 11, -12, -8, 29, -7], [-20, 22, -21, -16, 16, -3], [2, 35, -29, -17, 12, -2], [-7, 6, 5, -9, 1, -10], [-16, -3, -8, 14, -3, 3], [11, -4, -6, 7, 11, -15], [19, 21, 3, 1, 18, -11], [-14, 25, -10, -32, -11, 7], [-24, -3, -17, -24, -7, -12], [-40, -17, 18, 6, -7, 7], [-12, 11, 15, 13, 22, -26], [7, -2, -11, 12, 32, -8], [15, 10, 17, 15, 9, 8], [-28, 16, -33, -20, 6, -17], [26, -7, -19, 13, 20, -16], [-26, -19, 20, -9, -10, 1], [-10, -5, -17, -2, 1, 2], [17, -21, 7, 26, 1, -25], [12, 13, -15, -24, 23, 4], [2, -36, 15, 4, -22, 3], [-36, 4, -3, -15, 3, -7], [7, 1, -29, 2, 11, 11], [18, 0, -13, 12, 6, 19], [-16, -9, 31, 12, -19, 0], [20, 0, -12, -22, -16, 28], [16, -2, -5, 1, -1, -13], [-17, -6, -3, 5, 15, -4], [12, -2, -5, 14, -14, -1], [-16, 10, -4, 16, 4, -22], [-41, -8, -1, -11, -3, -12], [-16, 20, 22, 18, 6, -12], [-3, -10, -16, 21, 1, -3], [-8, 18, -14, -28, -6, 24], [5, 0, -28, -7, -5, 13], [53, 5, -20, 2, 11, -2], [10, 14, 1, -23, 3, 11], [-17, 3, 20, 13, -6, -10], [-20, -9, 15, 32, -11, -6], [18, 7, -1, 1, 30, -6], [-11, -29, 16, 4, -13, 22]];
    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_m();".
// 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_m(:prec:=6)
    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_m();".
// 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_m( : 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, 10, -3, -16, -4, 3, 1]>,<5,R![-32, 40, 38, -25, -12, 3, 1]>,<17,R![296, 596, -98, -253, -48, 5, 1]>],Snew);
    return Vf;
end function;