
// Make newform 2673.2.a.p in Magma, downloaded from the LMFDB on 20 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2673_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_2673_2_a_p();".
// 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_2673_2_a_p();".
// 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 := [-9, -16, 14, 14, -8, -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, 0, 1, 0, 0, 0], [-1, -5, 0, 1, 0, 0], [7, 5, -7, -1, 1, 0], [0, 11, 5, -8, -1, 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_2673_a();"
function MakeCharacter_2673_a()
    N := 2673;
    order := 1;
    char_gens := [2432, 244];
    v := [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_2673_a_Hecke(Kf)
    return MakeCharacter_2673_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, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, -1, 0], [1, 0, 0, 0, 0, 0], [-2, 0, 0, -1, 0, -1], [1, 0, 0, 0, -1, 1], [-1, 2, 0, 0, 0, 1], [2, 0, 1, 0, -2, 0], [1, 0, 0, 0, -1, -1], [-1, 0, 1, 0, 0, 2], [-1, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2], [1, -2, 2, 1, -1, 0], [4, 0, 0, -1, 1, 0], [4, -2, 0, -1, 1, 0], [-2, -2, 0, -2, 0, -2], [0, 0, 0, -1, 1, 1], [2, 0, 0, -1, -1, 0], [4, -2, 1, -1, 1, -2], [2, 0, 2, -2, -2, 0], [-1, -2, 0, 0, 0, 1], [-1, 4, -2, 1, 2, 0], [0, -2, -1, 0, 0, 0], [-3, -4, 2, -1, 1, 0], [2, 4, -2, -1, -3, 1], [6, -4, -1, 0, 2, 0], [12, 0, -2, 1, 1, 1], [1, -2, -2, 0, -2, 1], [-8, -2, 2, -1, 1, 2], [-9, 2, -2, 0, 2, -1], [-3, -2, 0, -3, 0, -2], [2, -2, 0, 2, 0, 0], [4, 0, 0, 3, -2, -1], [4, 0, -4, 3, -1, -1], [0, 0, 2, 1, -3, 1], [-1, -4, 0, 0, 0, 2], [5, -4, 1, 2, 2, -2], [4, -2, 2, 2, -2, 2], [-3, 2, -4, -2, 1, 1], [-4, 6, 2, -1, -3, 0], [1, 4, -3, 3, 1, 0], [0, -2, 2, 0, 0, 4], [3, -2, -4, 1, 0, 0], [5, -2, 4, 1, -4, 2], [5, 0, 6, -1, -1, 2], [-7, -6, 6, 0, 0, -1], [-7, 4, -3, 1, 1, 2], [0, -2, 2, 1, 1, 3], [-1, -4, 2, 3, 3, 0], [13, 4, 0, 1, 0, 0], [9, -4, -2, 1, 4, 0], [1, -2, 6, -1, -3, 0], [-4, 2, 3, -1, -3, 2], [4, 2, -1, -1, 1, 2], [11, -6, -2, 1, 2, 0], [8, 2, -5, 3, 1, 0], [-2, -4, 4, -1, -3, 1], [-5, 2, 2, -3, 1, 0], [0, -2, 2, -1, -1, -3], [-5, -2, 2, -2, 2, -1], [9, 0, 4, -2, 1, 1], [2, 4, -2, 3, -3, -1], [4, 2, 5, -1, 1, 0], [7, 0, 1, 3, 1, 2], [-8, -2, -2, 2, -2, 0], [-7, 0, -4, -2, 4, 0], [-2, -4, 4, -3, -2, 1], [8, 2, -2, -2, 2, 0], [-7, 2, 0, -1, -1, -2], [-4, -8, 7, 3, 1, 0], [1, 6, -2, 2, 1, -1], [12, 4, -3, 3, 5, 4], [-4, 0, 0, 5, 2, 1], [1, -8, -2, 0, -2, -2], [8, 4, 0, 1, -1, 6], [-2, 4, 6, 1, -3, 0], [5, -4, 0, -4, 2, -6], [2, -2, -2, 0, -2, 4], [-15, 2, -4, 0, 2, -1], [10, -4, 1, -4, -2, 0], [3, 12, -2, 3, -1, 4], [14, 2, 2, -1, -3, 5], [-9, 4, -1, 1, 3, 0], [1, 2, -8, 3, -2, 2], [-4, 14, 0, -2, 2, 2], [-6, -4, -1, 1, 1, -4], [11, -6, -2, 1, 1, 6], [-4, -6, 2, 4, -4, 0], [7, 8, -5, -5, 5, -4], [-4, -6, 4, 3, 1, 2], [-10, 6, -6, -2, 8, 2], [5, 4, 7, 0, -6, 2], [1, 4, 4, 1, 0, 0], [22, -4, -1, 2, 0, 2], [-3, -8, 4, -6, -3, -5], [4, -6, -4, 2, -2, -2], [6, 0, 4, 0, -6, -2], [-8, 0, -2, -1, 3, -1], [10, 4, -6, 2, 3, 0], [2, 4, -2, -1, 5, -7], [2, -4, 4, -3, 7, 3], [11, -4, -2, 0, 1, -3], [-3, 2, 2, 3, 6, 4], [-10, 0, 8, -4, -1, -4], [-2, 12, -1, 1, -1, 2], [-6, -2, -5, 1, 1, -2], [8, 0, 6, 2, -6, 2], [4, 6, -8, 4, 0, 2], [-13, -2, 6, -6, 0, 1], [-10, 0, 6, 3, 0, 3], [0, -8, 2, -1, 1, -1], [12, -4, 2, -4, -4, -2], [0, 0, 4, 2, -2, 0], [-1, -4, -1, -1, 5, -4], [-18, 4, -1, -3, -3, -6], [6, 4, 2, 0, 2, 2], [18, -2, -1, 4, -2, -2], [-10, 8, 0, 4, 2, 2], [-24, 0, 2, 1, 1, 1], [6, 4, -2, 1, 3, -4], [-8, -4, -2, -1, 0, 1], [-12, 10, 2, -6, -6, 0], [6, -10, 5, 2, -4, 2], [-7, 4, 2, -1, -1, 0], [12, -4, 0, 1, 1, 3], [21, -4, 4, 2, -2, 8], [2, 10, -1, -2, -4, 4], [-6, -8, 4, -3, -7, 0], [-11, 6, 0, 0, -2, 9], [-1, 2, -6, -6, 0, -1], [14, -16, 0, 2, 0, 2], [-13, -4, -8, -3, 3, -8], [10, 8, -5, -4, 0, -2], [-22, -8, 2, 0, -2, -2], [-15, -10, 6, -1, -5, -2], [2, 14, -5, -4, 6, -2], [1, 22, 2, 0, -5, 5], [-18, 8, -10, -4, 2, -4], [-4, 0, -2, 2, -6, 2], [-7, 10, -8, -3, 3, 0], [-6, -6, 6, 2, 8, -4], [-15, 4, -4, -2, -6, -2], [-12, 10, 2, 3, 3, 1], [-7, 4, 6, -3, 3, 2], [12, 0, 8, 1, -5, -2], [-5, -6, 2, 0, -5, 3], [-7, 0, -4, 3, -2, 4], [1, 4, 1, 4, -4, 8], [18, 0, 5, 1, 1, -4], [12, -8, 6, 2, -5, 2], [24, -14, -1, 4, 4, -2], [1, -8, -5, 7, -1, -2], [1, 0, 8, -8, 3, -1], [-3, 12, 8, -2, -6, 0], [6, -2, 9, 1, 1, 4], [7, -6, 6, -2, -1, -7], [4, 6, -6, -1, 1, 2], [12, -4, 6, -1, 4, 1], [-10, 12, 0, -5, -7, 1], [10, -8, -2, -1, 7, -2], [-1, 14, -4, 0, -11, 7], [-1, 14, -8, -1, -4, 4], [14, 16, -2, 2, 0, 4], [-8, 8, 2, -2, -6, -6], [4, 2, -10, -5, -3, -6], [2, 0, -6, 3, 3, 4], [-21, 10, -2, 2, 1, -5], [6, -12, 2, -1, -5, -5], [-12, -2, 6, 4, 4, 4], [-1, -6, 12, -5, -4, -4], [-8, -12, -3, -2, 8, 2], [19, -14, 0, -2, -9, -5], [-8, -4, 2, 0, 7, -4], [-3, -12, -8, -1, 7, 0], [-8, -8, 3, -4, 4, -8], [-1, 6, -2, -2, -2, -5], [-2, -12, -1, -5, 3, -10], [-11, 2, -6, 2, 0, 3], [13, 6, 4, 2, -6, 3], [5, 12, -2, 6, 0, 4], [-7, 4, 4, 1, 8, 4], [18, 12, -6, -1, -11, 4], [-5, -4, 6, 0, 5, 1], [18, 0, -4, 5, 5, -4], [-2, 0, -8, -3, 5, 0], [-7, -6, 14, 3, 0, -2], [11, 8, -2, -6, 0, 2], [-17, 10, 0, 6, -2, -1], [-2, 4, 10, 3, -7, -1], [15, -12, 1, 3, -1, 6], [10, 4, -12, 1, -9, -7], [13, 0, -1, 2, -6, 2], [-14, -2, 7, 2, 4, -2], [-1, -4, 10, -2, -1, -5], [-18, 10, -3, 1, 1, 4], [-18, -4, 8, 5, 4, 3], [15, -12, 14, 4, -6, 2], [12, 6, -14, 3, 3, 1], [-11, -10, 10, -5, -6, -6], [-15, -2, 6, -5, -2, 4], [-14, 12, 6, 2, 3, 0], [19, 4, -5, 8, -6, -4], [3, 10, -2, 2, 13, -1], [-16, 12, -3, 3, 1, 2], [-10, -4, -10, 1, 5, -8], [-11, -4, -12, 0, 4, -4], [3, -4, 6, 4, 1, 5], [-4, 4, -6, 2, 6, -4], [11, -12, -2, -9, 9, -4], [5, -6, 6, -6, 6, -7], [-16, -4, 8, 4, 8, 2], [18, 0, -5, -3, 5, -4], [-2, -10, -4, 1, -3, -4], [7, 2, -12, 5, -2, 4], [-10, 4, 2, -9, -3, 6], [2, 4, -12, 3, 0, -5], [-12, 14, -8, 0, 12, 0], [18, 14, -13, -4, 2, -2], [4, 0, -4, -5, -3, -2], [4, -4, 6, 1, 8, -5], [-29, 2, 2, -2, -1, 1], [-26, -2, -8, -2, 0, -8], [7, -16, -4, 0, -2, -10], [11, -18, -2, -2, 11, -1], [-15, 14, -8, -4, -8, -1], [26, 0, 8, 5, -3, 12], [-22, 14, -4, -12, -2, -4], [-1, 6, -6, 5, -7, -2], [-12, 4, 0, 5, 5, -3], [-17, 4, -9, -2, 8, -6], [-16, -24, -4, 0, 9, -6], [0, 20, -8, -2, -9, 6], [-26, 16, 8, 1, -9, -3], [-9, -8, 13, -5, -9, -4], [39, 10, -6, 3, 0, 0], [3, -8, -8, -2, 0, 6], [10, -22, 2, 1, 3, 3], [10, 16, 4, -3, -7, 6], [2, 10, -4, -1, -9, 0], [0, 16, 1, -1, 5, 4], [35, 6, -6, 1, -2, -6], [14, 0, 1, -1, -1, 6], [-4, 12, 1, -2, -8, 0], [-34, 4, 4, 4, 2, 2], [8, -10, -2, -9, 7, -5], [-5, 8, -2, -7, -3, -2], [0, 12, -8, -2, 10, -4], [15, -2, 2, -5, -3, 4], [15, -4, -6, 6, 3, 1], [-18, 12, 8, -1, -5, -1], [24, 4, 12, -5, -5, 5], [16, -6, 1, 6, 2, -6], [-8, 0, -10, 1, -7, 6], [4, 4, -4, 7, 3, 5], [-6, -16, 4, -4, 2, 2], [-20, 0, 3, -1, 3, -4], [7, -2, -4, 1, -7, 12], [-6, 2, -10, 7, 7, -2], [-12, 12, -8, 0, 5, -2], [-8, -24, 0, 0, 13, -4], [-4, 2, -12, 3, -5, 3], [18, -12, 0, -7, 1, -7], [-11, -6, 8, -3, -11, -4], [20, -12, -1, -5, -7, 6], [11, -4, -6, -3, 15, 0], [2, -6, 6, -1, 3, 10], [-30, 0, -6, -3, 9, 0], [-22, 12, 10, 2, -7, -2], [-1, -4, -6, 10, -1, 3], [-3, -4, 3, -4, -2, 4], [-5, 12, -13, -3, 1, -4], [15, -16, 0, 2, -8, -4], [-9, -6, 12, -2, 3, 11], [35, 6, -12, 0, 0, 3], [-34, 4, 0, -2, -8, -8], [24, -6, 0, -5, 1, 0], [-11, -10, 18, -2, 2, -5], [2, 4, 12, -3, 0, -7], [31, 2, 10, 0, -7, 11], [-11, 2, 4, -8, 3, -5], [21, 12, -10, 9, -1, 4], [6, 18, 4, 0, -6, 2], [11, -2, -4, 3, 6, -6], [13, -10, 12, 6, 1, -5], [-18, 6, -5, 1, 1, 4], [15, 0, -5, 1, -3, 8], [-24, -2, 0, -2, -2, -2], [-16, 8, -5, 0, 0, -2], [-17, -10, -10, -3, 2, -10], [26, -10, 2, -3, -11, 0], [-24, 16, 4, 5, -1, 6], [24, -4, -6, 0, 0, -4], [2, 8, 2, 7, -7, 5], [-28, -16, -6, -1, 5, -5], [-8, -6, 2, 4, 0, 8], [7, 0, -13, -2, 2, 4], [-14, 2, 12, 2, 4, -6], [-34, 6, 0, -6, -8, -8], [8, -12, 2, 4, -5, 2], [-25, -18, 2, 0, 6, 1], [0, -14, 16, -1, -1, -1], [-12, -8, -2, 0, -1, -4], [30, 0, -1, -1, -1, 4], [-19, 0, 1, -4, -16, -8], [-11, 16, 7, -1, -3, 4], [28, -12, 13, 6, 2, 0], [6, -16, -2, 10, -6, -2], [6, 16, 11, 0, -6, 0], [-24, 8, 4, 0, 5, 10], [14, -22, -8, 5, 3, -5], [-10, -18, 18, -4, -6, -6], [-8, 6, -10, 4, -12, -8], [-1, 0, -3, -2, 10, -12], [7, -20, 4, 7, 0, -2], [-37, 6, 10, -2, -2, 1], [-8, 4, -2, -4, 4, -16], [-18, 8, 2, 6, 12, 0], [-5, -4, -2, 1, 5, 8], [30, 12, -4, -2, 4, -6], [-14, 12, -6, -3, -5, 11], [7, -12, -8, 2, -6, -6], [42, 16, -6, 4, -2, 2], [-17, -4, 5, -1, 9, -4], [14, 6, 12, 5, -9, -3], [31, -12, 4, -8, 2, -4], [14, 0, -12, -7, 5, 2], [2, -20, 4, -2, -8, -4], [-1, 4, 2, 2, 15, -11], [-24, -8, -24, 7, 7, -3], [30, -4, 2, 1, -15, -5], [-2, 12, -16, 9, 5, -2], [-15, 10, -6, 1, 9, 0], [3, -2, 8, 0, -3, 11], [-14, 0, -6, 4, -10, 0], [20, -8, 2, -4, 12, -8], [-12, -8, -4, 5, 1, 6], [-42, 4, -4, 5, 11, -1], [7, 4, -4, 0, -2, -10], [-4, 16, 10, -3, -3, 5], [27, 4, -14, -3, -1, 4], [0, 2, -8, -8, 4, 0], [12, -2, 8, 1, -11, 9], [23, -4, -6, -2, -2, -10], [-10, 14, 10, -9, -5, -4], [-4, 10, 0, 5, -3, -11], [18, -4, 9, 1, 3, -10], [21, -20, -4, -1, -5, -10], [24, -10, -4, -3, 9, 7], [-22, -8, 12, 8, -6, 2], [56, -16, -4, 2, -1, -2], [-19, 8, 9, 8, -4, -2], [-10, -12, 5, 1, -7, -6], [-29, 2, -10, -6, -5, -11], [-6, -16, 6, 2, -2, 2], [-39, 12, -4, 1, 4, 8], [14, -22, -2, -4, 6, 10], [-28, -24, 4, 1, -7, 3], [9, 0, -20, 4, -5, -1], [32, -16, 2, 3, 1, 0], [9, 14, -16, 5, 1, 0], [-15, 24, -6, 5, -4, 18], [-20, -2, 9, 3, 17, 0], [-35, 10, -4, 4, 3, 7], [-11, 4, 6, 0, -14, -8], [-5, 8, 8, 9, -5, -6], [-47, -12, 0, 1, 0, 2], [-21, -2, 0, 9, 23, -6], [-18, -4, -6, -2, -3, -6], [-30, 24, 4, -4, 2, 10], [-13, 4, -3, 5, 5, 0], [32, -8, -6, 5, -3, 1], [4, 14, -4, 8, -8, -4], [6, 4, 4, 2, 1, 2], [5, 8, 1, 7, 1, 4], [12, 12, 6, 3, -3, 10], [30, -8, -10, 6, 0, -2], [16, -4, 4, 11, 0, 13], [-16, -16, 18, -11, -7, -9], [7, 14, 4, -3, 4, -12], [-26, -22, -5, 5, 13, -8], [-4, 20, -21, -5, 5, -6], [-7, -2, 2, -2, 10, 9], [2, -26, 8, -1, 9, -7], [-27, -26, -2, -5, 9, -10], [8, -20, -4, -3, 7, 4], [6, -4, -8, -4, -14, -8], [-27, -8, -3, -4, 4, -6], [9, -18, -4, -6, 2, -11], [-10, 24, -5, 2, 12, 0], [35, -12, 11, -2, -8, 2], [0, 8, -18, 2, 2, 4], [-14, 20, -4, -5, 3, 10], [-37, -4, 4, -7, -1, -8], [0, 26, 1, -3, 3, 0], [7, -6, 10, -8, -1, -3], [11, -2, 16, -1, 5, 8], [-14, 16, -2, 1, -3, 10], [-7, 16, -7, 3, 21, 4], [-34, 10, 4, -4, -6, -8], [32, -4, -10, 4, 1, 4], [2, -4, 6, -1, -9, 2], [-28, 12, -10, 1, 4, -5], [10, -22, 9, 4, -6, 4], [-28, 6, 10, 0, 4, -8], [-23, -26, 6, 3, 1, 4], [-11, -20, 14, -5, 5, 6], [-20, -6, 12, -7, 1, 7], [-16, -4, 5, 3, -15, 4], [-36, -14, -15, 5, 11, -10], [28, -12, 4, 6, 7, -8], [7, -4, 8, -3, -10, -10], [-40, 16, -9, -1, 9, -2], [11, 14, 8, -5, -8, 10], [19, 26, -2, 3, -5, 4], [4, 0, 7, 6, 2, -6], [24, -6, 6, 7, -5, 13], [39, -8, 7, -6, -4, -6], [14, 8, -4, 5, 9, -6], [-40, -12, 10, -4, 0, -4], [-12, 20, 4, 0, 12, -2], [-3, 18, -2, -8, 0, -5], [22, 18, -10, -2, 0, -2], [13, 8, 6, -7, -8, 0], [-11, -16, 0, -1, 15, -8], [38, 0, -12, 9, 13, 10], [-8, 2, -9, -10, -2, -8], [-16, -10, -2, -5, -1, -15], [-18, 4, 14, -2, -12, 0], [-12, -8, -4, -15, 9, -13]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2673_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2673_2_a_p();".
// 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_2673_2_a_p(:prec:=6)
    chi := MakeCharacter_2673_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_2673_2_a_p();".
// 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_2673_2_a_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2673_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-9, -16, 14, 14, -8, -2, 1]>,<5,R![-48, 8, 77, 14, -17, -2, 1]>],Snew);
    return Vf;
end function;