
// Make newform 2775.2.a.be in Magma, downloaded from the LMFDB on 28 January 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2775_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_2775_2_a_be();".
// 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_2775_2_a_be();".
// 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 := [-6, 2, 23, 7, -9, -2, 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], [-3, -2, 1, 0, 0, 0], [6, 0, -14, 2, 4, -1], [-3, 8, 10, -5, -3, 1], [5, -12, -12, 6, 3, -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_2775_a();"
function MakeCharacter_2775_a()
    N := 2775;
    order := 1;
    char_gens := [926, 1777, 76];
    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_2775_a_Hecke(Kf)
    return MakeCharacter_2775_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], [-1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [-1, 0, 0, 1, -1, 0], [0, 0, 0, 0, -1, -1], [-3, 0, 0, 0, 0, 1], [1, 1, 1, -1, -1, 0], [-1, -1, 0, -1, 1, -1], [-1, -2, -1, 0, 1, 1], [3, -2, -2, -2, 1, 0], [1, 0, 0, 0, 0, 0], [0, 0, -1, -1, 1, -1], [0, 1, 0, -1, 0, -1], [-1, -4, 0, -1, 1, 0], [-7, 0, 0, -1, 0, 0], [0, -1, 0, 0, 2, 1], [0, 0, -3, -1, 1, 2], [1, -4, -3, -1, 3, 1], [-1, 4, 0, 2, 0, 1], [-3, 2, 1, 1, 1, 2], [-1, 1, -2, -2, 0, 2], [-7, 2, -1, -1, -2, 2], [-1, -2, 0, 2, 0, -1], [4, -4, 1, 0, -2, -1], [-4, 4, 4, 3, 0, -1], [-1, 1, 3, 0, 1, 0], [1, -2, -1, -2, 5, 1], [4, -2, -1, -2, 2, -1], [-5, 2, 0, -2, -2, -1], [-3, -4, 2, 1, -3, 2], [-5, 0, -3, -2, -1, 1], [-3, 2, 3, 1, -6, 0], [4, 2, 3, 3, 2, -1], [-2, 4, 3, 6, 0, 0], [-3, 0, -4, 0, -3, 2], [0, -4, -1, 2, 3, 1], [6, -1, 3, 2, 2, -2], [-4, 5, 3, 3, 0, 2], [1, -6, -1, -1, 4, -2], [-3, -2, 1, -5, -2, -2], [-3, -8, -3, -5, 2, 1], [-10, -1, 0, -1, -1, -2], [-9, 2, 0, 0, 1, 4], [3, 4, 2, 0, -3, -3], [-4, 7, 1, 0, -4, -2], [2, 0, 4, 0, -3, -3], [-6, 2, -5, -1, -1, -2], [-1, -1, 4, 2, -2, -2], [1, 6, -2, 1, 3, 1], [-7, -4, 1, 1, -1, 2], [6, 5, 6, -3, -3, -2], [0, 4, 0, 0, -1, -1], [4, 1, -1, 0, 1, -5], [-14, 2, 0, 1, -3, 1], [5, -2, 3, 2, -3, 1], [0, 4, 6, 0, -1, -2], [9, 4, 3, 3, 0, -4], [-1, 4, -1, 1, -5, -1], [-5, -2, -4, -3, 1, -2], [6, -1, -1, 0, 8, 0], [-9, 6, -1, 0, -2, -3], [-3, 0, -1, 0, 0, 2], [-11, 7, 7, 3, -2, 1], [-13, 0, 0, 2, 1, 0], [-1, -2, -1, 1, 1, -2], [0, -2, -1, 3, 4, 3], [-9, -8, -2, -5, 3, -1], [-6, 1, 0, -7, -5, -2], [0, 2, 4, 3, 2, 5], [4, -2, -2, -1, 3, 1], [4, -2, -2, -6, 6, 4], [2, 8, 6, 6, -1, -1], [-3, -2, -2, 2, 1, 2], [7, 0, -3, 4, -4, -6], [8, -7, -3, -3, -2, 2], [2, -6, 5, -4, 1, 2], [-8, 0, -2, 1, 5, 6], [1, -4, 2, 1, -1, 4], [9, 12, 4, 8, -5, -2], [3, 2, -3, 0, 7, -1], [-9, 4, 5, 9, 0, 2], [-11, 9, 8, 4, -4, -2], [4, -2, 2, 0, -1, -3], [7, -3, 0, -2, 1, -3], [9, -6, -4, 4, 6, 3], [11, -8, 1, -1, 2, -6], [-17, -4, 0, -1, 1, 0], [1, 8, 4, 6, 0, 1], [-3, 2, -2, -3, -3, 0], [5, 6, 1, 1, 2, -2], [5, -6, 0, 1, -3, 2], [-1, 6, 5, -2, -2, -6], [-5, 0, 2, -1, -7, -6], [-11, 3, -3, 2, -4, 3], [1, 12, 8, -2, -4, -3], [-8, 0, 2, 0, -7, 2], [2, 0, 8, -3, 0, -5], [5, -8, -2, -1, -4, 1], [-5, 0, 3, 8, -4, 4], [-4, 6, 4, -7, -1, 1], [-15, 0, -4, -9, -5, 0], [8, 3, 4, 3, -1, 2], [-1, -12, -2, 3, 5, -4], [1, 0, 0, -7, -2, 5], [-5, 0, 0, -6, -5, -4], [-3, -7, -7, -2, 1, 0], [-13, 4, -6, -2, -3, -5], [13, -8, -2, -1, 9, -6], [-9, 4, 3, 6, 1, -2], [4, 2, -10, -1, 3, 3], [-3, 12, 2, -2, -7, -1], [0, 2, 7, 5, -6, -3], [25, -6, -1, 2, 0, 2], [17, -14, -9, -4, 8, 4], [-13, -2, -6, -1, 0, 2], [-2, 8, 4, 8, -2, -2], [-7, -1, 3, 8, -7, 0], [-14, 0, -1, -4, -7, 0], [4, 0, 3, 3, -2, 5], [0, -8, 0, -5, -5, -1], [-7, -4, -6, 3, 8, 2], [34, -2, 1, -3, -3, 1], [6, 12, 2, 8, -2, 0], [12, -12, -3, 1, 2, -1], [-13, -2, 4, 4, -2, 1], [-12, 14, 7, 2, 2, -1], [8, -6, 0, -3, -3, -1], [1, -2, -2, 10, -3, 2], [3, -2, -3, 1, 2, 4], [9, -16, -9, 1, 4, 2], [-4, -2, -7, -3, 0, 3], [-2, 0, 6, -3, -3, -3], [11, 0, -4, 4, 7, 4], [9, -18, -10, -3, 10, 0], [3, 0, -5, -3, 9, 1], [-7, -6, 0, -3, 9, 2], [0, -8, 2, -6, -5, -7], [1, -4, -10, -3, 3, 0], [-3, -4, -6, 1, -3, -4], [20, 8, -2, 10, -1, 5], [-10, 10, 4, -2, -9, 2], [4, 14, 11, 8, -4, -1], [11, -10, -1, -1, 8, 0], [-17, -2, -4, 3, 1, -2], [-17, -8, -10, 6, 6, 3], [-15, 18, 10, 8, -11, -6], [11, -6, -4, 4, -8, 3], [20, -7, -7, -4, 3, -5], [18, -6, -2, 4, 0, 2], [1, 0, -1, 4, 1, -2], [-12, 18, 7, 5, -11, -5], [12, 14, 5, 0, -6, 1], [-7, 2, 1, 5, 0, -4], [-12, 9, -1, 1, 8, 6], [-3, -9, -1, -1, -2, -5], [2, -4, -4, -1, 14, 4], [10, 4, -6, 0, 6, 0], [-27, 10, 3, 4, 4, 0], [0, -4, 1, -4, -3, -10], [-13, 3, -4, 9, 3, -1], [-29, -10, -2, 3, 2, 2], [-9, 9, -7, 3, -6, 1], [-12, 4, 5, 12, -1, 0], [-6, 16, 6, 8, -2, -2], [-6, -8, -4, 0, 0, 6], [-6, -3, 1, -1, -7, 1], [5, 2, 8, -5, -3, -4], [3, -8, -11, -7, 6, 1], [-13, 0, 14, 2, 0, 1], [-7, 2, 4, 3, -11, -10], [2, 10, 6, 2, -5, 7], [-15, -4, -7, 4, 1, 9], [-25, 18, 1, 5, 4, -1], [11, -4, 6, -7, 4, 7], [-1, 6, -2, 8, 4, -1], [-14, 11, 12, 6, -9, -2], [15, -18, -13, -2, 6, 3], [9, -6, -12, -1, 8, 7], [-3, -8, 1, -11, 3, 5], [-28, -5, 1, -3, -3, 3], [0, 1, -11, -5, 2, -10], [8, -8, -5, -3, 15, -2], [-2, -10, -13, -2, 5, 0], [30, -2, 6, 5, 11, -3], [-6, 0, 9, 4, -6, -2], [-18, 10, 3, -10, -5, -9], [17, 3, 6, 2, -1, 7], [-3, 8, 4, -12, 1, -2], [-12, 4, -12, 3, 1, 7], [3, 12, 15, 4, -8, -4], [-11, -10, -8, -9, 3, 8], [-20, -5, -9, 2, 8, 10], [12, 4, -6, -2, 0, 4], [11, -8, 5, -6, 1, 5], [-17, 16, 7, 5, -5, 0], [8, -16, 2, 4, -2, -2], [-2, -4, -4, 10, 2, -2], [-2, 8, 3, -4, -5, -8], [-10, 2, -16, -1, -7, 1], [7, -2, -1, -7, 14, -2], [9, 16, 0, 8, 0, 3], [8, -10, -9, -6, -6, 2], [-1, 4, 5, -12, 5, -5], [12, -4, -3, 6, 5, 0], [32, 6, 7, -4, -7, 2], [-30, -2, 1, -2, 2, 1], [-6, 18, 3, 5, -4, 1], [2, 16, 10, 4, -10, -4], [3, -14, 5, 1, -1, -5], [-2, 8, 6, -7, -2, -10], [-6, -10, 2, 0, 4, 8], [28, -6, 2, -2, 5, -5], [11, 4, -1, 3, -4, -6], [-2, 3, -7, 0, -1, -5], [16, 6, 7, 2, 4, 2], [1, 17, 6, 3, 2, -6], [9, -6, -11, -13, 4, 0], [-3, 13, 3, -12, -5, -4], [-7, -2, -1, 3, -4, -10], [29, 4, -11, 0, 1, 4], [10, 4, 5, 10, -10, -1], [14, -8, -2, 8, 2, -4], [9, -1, 4, -2, -12, -10], [9, -6, -16, -9, 1, -4], [-25, -16, 0, -8, 11, -6], [-3, -10, -10, 7, 10, 12], [-29, 2, 2, 2, 6, -3], [20, -12, 3, 1, 13, -2], [7, -13, 6, 11, 5, -5], [-21, 12, 0, -4, 0, 6], [22, 12, 4, 4, -2, 6], [-38, -2, -10, -3, 0, 2], [6, -4, -6, 1, 8, 7], [9, -20, 0, 5, 9, -2], [-6, 6, 5, 10, -1, 6], [-8, -12, -1, -2, 2, 4], [-3, 0, 6, -3, 7, 2], [18, -1, -8, 2, 4, -3], [28, -10, -6, -9, -3, 1], [-5, -11, 5, -5, 10, 1], [3, 0, -12, -3, 2, 3], [-11, -14, 5, 2, -6, 6], [-35, 16, 3, 3, -17, -11], [-31, 16, 8, 4, 1, 5], [3, -11, -9, -7, 0, 7], [-2, 0, -1, -9, 1, -4], [-27, 1, 11, 11, -6, 1], [-6, 26, 12, 13, -14, -8], [5, 5, 3, -5, 4, -9], [-11, 10, -1, 8, 7, 3], [-6, -4, -8, -14, -8, 4], [11, 12, -5, 2, 3, 5], [-19, 0, 9, 1, -11, -7], [0, 0, 7, -14, 3, -4], [13, -6, -13, 0, 9, 7], [32, -6, 2, 7, -5, -8], [17, -8, 2, 1, 5, 6], [2, 20, 6, -2, -4, 2], [-41, 0, 4, 7, -5, 8], [38, 2, 5, 10, 0, 1], [-16, 17, 4, 4, -8, 1], [19, -22, -9, -4, 6, 5], [-10, -4, -9, 2, 1, 11], [7, -4, -12, -10, 4, 1], [17, -2, -8, -12, 2, 1], [16, -8, -14, -6, 8, 8], [-11, 4, 3, -7, -2, 6], [18, -8, 9, -2, 9, -8], [-2, -4, 2, -12, 16, 2], [15, -10, -4, 5, 6, 7], [12, -3, 5, -5, -4, 8], [-2, 6, -8, 5, -9, 5], [-5, 1, -4, 3, 16, 4], [-24, 18, -1, -4, -5, 15], [5, 2, -8, -10, 7, 12], [13, 18, 14, 7, -13, 4], [-3, -6, -10, -1, 2, 3], [20, 2, -13, 0, -1, 5], [-28, 20, 3, 2, -13, 2], [-14, 2, -5, 4, -3, 0], [6, 13, 6, 5, 3, -4], [21, -8, 1, -1, 2, -16], [-20, 8, 13, 6, -21, -11], [33, -10, -11, 2, -10, 0], [3, 6, 11, -3, -2, -6], [17, 10, 10, 14, 5, -2], [-6, -18, -11, -18, 4, 0], [0, 15, -1, 1, -2, -8], [-8, 12, 7, 0, -3, -2], [16, -6, 4, 11, -5, 3], [30, -5, -4, 2, -8, -1], [-28, -4, -10, 8, 2, 2], [7, 2, -7, 1, -2, -8], [-13, 12, 4, -1, -4, -3], [19, 0, 10, -12, -12, -11], [-12, -2, 5, -5, -5, -8], [6, -2, -12, -2, 5, 0], [-9, -21, 6, 6, -2, 6], [11, 2, -1, 14, 5, 9], [8, 0, 7, -5, -2, -13], [4, -10, -16, -11, -6, -7], [11, 4, 10, -4, -3, -2], [-14, 19, 1, -6, 3, -5], [-10, 3, -4, 3, -19, 4], [-7, 8, 3, 10, -8, -12], [1, 1, 7, -11, 8, 1], [-1, 6, 6, -1, -16, 3], [-17, 2, 1, 1, -4, -8], [25, -22, -15, 1, 12, 10], [33, 6, 4, 2, 4, 3], [35, -2, -7, 6, 8, 5], [-9, 18, 3, 5, 2, 6], [-25, -6, -5, -1, 1, 9], [8, 2, -6, -9, -3, -11], [-23, -4, -3, -1, 9, 15], [22, -6, -2, -5, 11, 7], [-2, -4, -14, -4, 12, 10], [11, -12, -15, -12, -3, -10], [-17, 10, -1, -6, 5, 13], [-40, 2, 3, -1, 4, 3], [-15, -8, -13, 3, -6, 7], [35, 8, 6, 13, 5, 0], [7, 8, -8, -4, 1, -6], [3, -16, -11, -5, -7, 0], [9, -2, -6, -7, 3, 2], [-1, 20, 2, 0, -3, 0], [12, 2, 2, -6, 3, 5], [22, -7, -4, 6, -16, 1], [-19, -14, -4, -11, -3, 6], [22, -12, -2, -5, -8, 4], [21, 2, 5, -1, -2, -14], [-16, -1, 5, 6, 2, -6], [-7, -1, 5, -11, -2, -5], [16, -8, -18, -7, -2, -4], [-2, 14, 3, 7, -18, 3], [6, -14, -11, 12, 1, 7], [-15, -8, -2, -18, 1, -7], [-4, -8, 10, 5, 4, -6], [26, 4, 4, -3, 5, 5], [-8, 8, 9, -6, 7, -3], [27, -14, -1, 2, 16, 11], [3, 8, -6, -8, -2, -15], [-16, 7, 9, 5, -15, 5], [16, 12, -1, 4, -3, -16], [11, 19, -6, 4, 2, 8], [-17, -4, -1, -3, 6, 5], [27, 8, 3, 21, 1, 11], [-17, 14, 14, 1, -15, -2], [38, 4, 1, 14, -2, 4], [-23, 12, 10, 19, 2, -5], [-14, -12, -8, 1, 16, -5], [4, 6, -4, -14, 2, 12], [-27, 13, -4, -11, -12, 2], [-31, 4, 5, 5, -4, -7], [-12, 8, 4, -10, -8, -6], [-34, 7, -9, -4, -11, -5], [-26, -7, -9, -5, 0, -2], [-24, 2, -10, 3, 8, 9], [30, -15, 6, 2, 16, -5], [-7, 8, -2, -3, 4, 7], [-7, -10, 11, -1, -7, -6], [-3, -2, -1, 21, 0, -6], [-23, 21, 7, -13, -6, -7], [8, -18, 2, -7, 9, 9], [29, -2, -5, 8, -8, -2], [9, -12, -5, 14, -1, -9], [14, -15, -13, 4, -7, 5], [-12, -6, -7, -15, -3, 10], [-19, -1, 2, -5, 5, -1], [-2, 11, 11, -7, -5, 1], [38, -14, -6, 6, -12, -12], [-27, 14, 1, -1, 3, -6], [-27, 26, 20, 13, -13, 0], [-30, -2, 9, 6, -14, 11], [-39, -4, -2, -8, 2, 13], [-24, -10, -3, -10, 19, 0], [32, -18, -22, -10, 9, 3], [2, -4, -4, 7, -3, 5], [-30, -12, 12, 9, -8, 2], [16, -10, -16, -10, 2, 0], [-8, 28, 5, -1, -6, -11], [14, 9, 11, 11, -2, -2], [-14, -10, 4, 19, -4, 3], [33, 3, 3, -6, 7, -6], [9, 4, 2, -12, 2, 1], [-16, 6, -12, 5, 14, 18], [27, -14, -2, 2, 0, 7], [10, -23, -2, -6, 22, -5], [6, -5, -13, 3, 6, 6], [-38, 26, 0, -4, -13, 1], [-9, 8, -2, -11, -5, 4], [11, -7, 4, 6, 0, -12], [24, 14, 2, 2, 1, -17], [23, -14, 7, -2, 7, -3], [18, -4, 2, 5, -2, -2], [26, 11, -15, -12, 10, -6], [9, 0, -8, 4, 13, -3], [11, 24, 12, 3, -14, -5], [23, -16, -18, -21, 0, 2], [9, 2, 12, 16, -1, 11], [-31, 2, -2, 4, 14, -1], [13, 15, 17, -7, 7, -12], [30, 8, 7, 2, 11, -10], [-41, 6, 1, 6, 11, 3], [30, 6, -1, 4, 7, -4], [17, 20, 2, 1, -8, -3], [-11, -6, -3, -5, 11, 0], [16, -4, 1, 20, -2, -5], [-18, -2, -16, 6, 10, 0], [8, -8, -3, 1, 7, 14], [12, 8, 14, 3, -8, 9], [18, 4, -20, -8, 8, 4], [-2, 4, -4, -7, 0, 10], [7, -18, 4, -5, -10, -1], [9, -10, -5, -5, 14, 4], [-49, -14, 2, 5, -5, 10], [-22, 8, -2, 0, -17, -11], [27, 4, -15, -14, 12, 9], [8, 11, 2, 11, 9, 4], [-17, 6, 13, 7, -8, -12], [-18, -30, -14, -12, 15, 5], [2, -4, 5, -11, 6, 17], [-17, 20, 13, 8, -3, -7], [4, 4, 10, -8, -19, 2], [-21, 6, -5, 5, 11, -2], [-1, -2, 5, 3, 0, 10], [27, 16, 9, 17, 7, -6], [-41, 14, 14, 5, 0, 11], [0, 27, 8, -8, -14, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2775_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2775_2_a_be();".
// 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_2775_2_a_be(:prec:=6)
    chi := MakeCharacter_2775_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_2775_2_a_be();".
// 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_2775_2_a_be( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2775_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![16, 29, -15, -29, -4, 4, 1]>,<7,R![60, 140, 75, -22, -19, 1, 1]>],Snew);
    return Vf;
end function;