
// Make newform 2366.2.a.bh in Magma, downloaded from the LMFDB on 24 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2366_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_2366_2_a_bh();".
// 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_2366_2_a_bh();".
// 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, 2, 21, 12, -10, -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], [1, -9, -26, 7, 4, -1], [-13, -68, -17, 24, 3, -2], [-3, 37, 43, -26, -7, 3], [-24, 41, 74, -33, -11, 4]];
    Rf_basisdens := [1, 1, 5, 5, 5, 5];
    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_2366_a();"
function MakeCharacter_2366_a()
    N := 2366;
    order := 1;
    char_gens := [339, 2199];
    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_2366_a_Hecke( : Kf := Kf)
    return MakeCharacter_2366_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, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [-1, 1, 0, 0, 0, -1], [1, 0, 0, 0, 0, 0], [0, 0, 0, -1, -1, 0], [0, 0, 0, 0, 0, 0], [0, 1, -1, 0, 0, -1], [-1, -1, -1, 0, 2, -2], [-1, 1, -1, 0, 0, 1], [2, -1, 1, -1, 1, 1], [0, 0, 0, 1, -1, 0], [0, 1, -1, 2, 0, -1], [0, 1, 1, -1, 1, -1], [4, 2, 0, 1, -1, 0], [-2, 0, 0, 0, -2, -2], [4, -3, -1, -1, 3, 1], [1, -2, 1, 0, 0, 0], [6, -4, -1, -1, -1, 1], [2, -2, 0, 1, 3, 0], [1, -1, -1, -1, 3, 1], [-2, -3, 1, -1, -3, 3], [2, 2, 0, -1, -1, -2], [-1, 1, -1, 1, -3, 2], [-2, 2, 0, -2, 0, 0], [0, -4, 0, 0, 4, 2], [-3, 0, 2, 0, 0, 1], [-2, 2, 2, 0, -4, -2], [6, -2, 0, 0, -2, 4], [-4, -2, 0, 0, -2, 0], [4, -1, -3, 1, -1, 1], [-1, -1, 3, -1, 1, -1], [-2, -1, 4, 1, -1, 2], [0, 2, -2, 2, 1, -2], [11, -1, 1, 1, 1, 0], [-6, 1, 3, -2, 0, 1], [0, 1, 3, 0, -1, -1], [9, -2, 0, 0, -4, 3], [2, 0, 0, -3, -1, 4], [8, 4, 0, 3, 1, -2], [-4, 1, -2, 3, 3, -2], [2, -6, 0, 2, 0, 0], [3, -3, 0, -1, -1, 1], [-6, -2, 0, -1, 7, -2], [-1, 4, 2, 3, -1, -2], [4, 2, -2, 2, 2, -2], [8, -4, 2, -2, 4, 0], [0, -2, 0, 1, 7, 0], [-4, 2, 0, 2, 6, 2], [-8, -3, 0, 0, 2, -2], [-9, 7, 1, 0, -8, 2], [3, -3, -1, 3, 1, -1], [2, -5, -1, -1, 0, 1], [-12, 3, 1, 3, 3, -1], [3, 4, 1, 3, -5, 0], [-10, -5, -1, -2, 2, -1], [-6, -3, -3, -5, 4, 1], [-5, -4, -3, 1, 5, 2], [-6, 0, 2, -1, 1, 2], [8, 5, 1, 3, -5, 3], [-2, -1, 3, 0, 0, -3], [13, -2, 1, 0, 6, -2], [-3, 6, -2, -1, 1, 1], [5, -6, 1, 2, -2, 4], [4, -2, 4, -1, -5, 0], [-8, -4, 0, 0, 0, 0], [12, 1, 1, 2, 4, 3], [0, 2, -4, 3, -9, 0], [-8, -5, 1, -3, -5, 5], [-10, 0, 0, -3, 1, 2], [-1, 0, -3, -3, -1, 4], [-20, 5, 1, 1, -3, -1], [-4, 3, -3, 0, 9, 1], [10, 2, 0, 3, -11, 0], [4, 5, 3, -1, -7, 7], [6, -6, 4, -3, 3, -6], [12, 2, 2, 3, -9, 8], [10, -3, 3, -1, -1, 1], [0, 1, 2, -3, -9, 4], [-4, 1, 5, 0, -4, 7], [-12, 3, 1, 1, -7, 1], [-15, 1, -1, -1, 7, -4], [-8, -3, 1, 4, 8, 3], [12, 1, 3, 0, -9, 3], [0, -3, -3, -2, 2, -3], [14, 0, -2, 3, -1, 4], [-10, 4, -2, 0, 0, 0], [6, -5, -1, -2, 5, 5], [11, -4, 6, 1, 1, 6], [-4, 0, -1, 2, 2, 5], [11, 1, -1, -2, 0, -1], [-5, -2, -1, 2, 0, 6], [2, 0, -4, 5, 5, 0], [20, -1, -3, 0, -3, -3], [8, 4, 4, 0, -12, 4], [6, 2, -2, -3, 5, -2], [6, -4, 2, 3, 3, 0], [8, 4, 1, 2, -2, 3], [-12, 5, -1, 0, 2, -9], [-6, -1, 0, -1, 1, 2], [-6, 1, -1, -2, 4, 3], [-4, -10, 6, -3, 1, 6], [14, 3, 3, -4, -2, -5], [-1, -1, -3, -1, 3, 4], [-6, 3, -5, 0, -3, -9], [0, 6, -2, 4, -2, 0], [8, -7, -1, -3, 7, 3], [10, -7, -6, -1, 3, -6], [-6, 1, -3, 3, 1, -3], [8, -9, 1, 0, 9, 7], [8, -1, -1, -2, 8, -1], [6, -4, 2, -2, 2, 8], [16, 3, -7, 0, 8, -3], [3, 2, 0, 5, -7, 0], [-17, -2, -1, -4, 0, -8], [3, 1, -7, 3, -1, -5], [2, 2, 0, -1, -2, 0], [21, 2, -3, 3, 5, -4], [-10, -8, -4, 2, 4, -4], [-14, 6, 4, 2, -8, -8], [8, 2, 6, -4, -6, 2], [-12, 12, 1, 4, -4, -5], [6, 7, -3, -1, -7, -7], [-11, -6, -3, 1, 5, 0], [12, -2, 0, 2, 14, -2], [-11, 2, -3, 3, 5, 4], [0, 10, 0, 2, 10, -10], [0, -5, 1, 0, 2, 7], [-4, -6, 2, 3, -7, 4], [0, 0, 6, 2, 0, 2], [-12, 4, -5, 2, 6, -11], [12, 2, 0, 2, -2, -2], [20, -6, -2, 0, -6, 10], [11, 1, -5, -2, -2, -7], [12, 4, 2, -8, -2, 6], [-18, -10, -2, 2, 6, 0], [16, -8, 2, -2, 12, 0], [-13, -5, 1, -4, 8, 2], [-8, 9, 0, 4, -10, -2], [16, -9, -2, -3, -3, -2], [0, -9, -3, 5, 3, 1], [4, -5, 4, 4, -2, 6], [-12, -6, 0, 0, -8, -2], [12, 3, 1, -2, -1, -1], [6, -6, -6, -7, 1, 4], [10, -4, 5, -7, 5, 7], [20, -6, 2, 0, 10, 2], [-5, -3, 0, 4, -4, -5], [-2, 5, 1, 0, 12, 1], [-1, 11, 1, -3, -3, -4], [-1, 5, 3, 0, -2, -5], [8, 3, 3, 2, -8, 11], [-8, 7, 1, 6, -10, -7], [-12, 4, -8, 3, -3, 2], [12, -12, -6, 2, 0, -2], [-6, -2, 0, 1, 9, -6], [9, 1, 5, 3, 5, -11], [12, 3, -3, 4, 17, -9], [-2, -1, 3, 1, 3, 5], [6, -1, -9, 2, -6, -3], [13, -2, -3, -5, -7, 2], [10, 6, 0, 1, -5, 4], [-7, 11, -3, -3, -1, -7], [18, 4, 2, -1, 7, 8], [-6, -9, -4, -1, 9, 6], [4, -4, 0, -8, 1, 0], [8, 10, -2, 6, 0, 10], [-2, 11, -1, -1, 6, -5], [2, 10, -5, 7, -9, -3], [25, -1, 5, 3, -1, 3], [3, -1, 6, -3, 17, -1], [10, 0, 6, -1, -13, 2], [-1, -5, 2, -4, -16, 9], [4, -5, -7, 0, 1, 7], [4, 6, -2, 2, 5, 6], [14, -12, -2, 1, 9, -6], [20, 3, 1, -2, -2, 3], [-2, 16, 6, 0, -12, -4], [12, -11, -1, 4, -10, 3], [-6, 12, 4, -6, 0, 6], [8, -11, 2, 3, -13, -2], [6, 5, 5, -3, -4, 11], [13, 1, 1, -4, -6, -2], [4, 3, 5, -7, 1, -1], [-2, -4, 4, 2, 4, -2], [28, -5, 5, 2, 3, 17], [-6, -8, 3, -5, 7, -11], [-12, 0, -2, -4, 6, -2], [13, 2, 5, -1, -7, 0], [-4, 2, -6, 4, 3, 2], [-12, 0, 0, 0, -12, 12], [-2, 3, -9, 0, 14, -9], [6, -6, -8, -1, 5, -6], [-2, 10, -2, 10, -2, -2], [-5, -7, 3, -4, -16, 0], [8, -10, -2, -1, -9, 10], [-30, 8, 6, 2, -6, 4], [-12, 1, 5, 2, 2, -5], [-6, -7, 5, -3, 1, -1], [-6, -5, -3, -7, -9, 3], [24, 2, 2, 2, -8, 16], [-8, -5, -9, 4, 16, -5], [16, 2, 4, -6, -14, 12], [11, -4, 8, 1, 3, -3], [16, 11, -1, -2, 2, -7], [25, -3, -7, 2, 0, -2], [-13, 17, -4, 1, -15, -11], [26, -4, -2, 5, -1, -2], [14, -4, 2, 5, -1, 4], [-5, -4, 2, -5, -17, 6], [9, 8, -1, -1, 11, -12], [14, 0, 6, 9, -2, 2], [-1, 1, -9, -4, 8, -4], [0, 0, 4, -6, -6, 8], [-6, -1, -2, -5, -13, 2], [-4, -2, -6, 8, 6, -6], [26, -19, -1, -1, 5, 9], [2, 5, 9, -1, -16, 1], [-16, 3, -5, -4, 12, -1], [-15, 3, -9, -5, 3, 1], [-14, 6, 8, 0, 6, -8], [2, -7, 1, 3, -7, 13], [2, 4, -10, 8, -4, -4], [2, -7, 1, 4, -4, 11], [0, 18, -2, -6, -8, -6], [-18, -5, 0, -9, -3, 6], [16, 11, 1, 3, 5, 7], [12, 5, -1, 0, 4, 1], [5, -3, -7, -3, 7, -11], [-2, 6, 2, 2, 10, -12], [16, -10, 2, 2, 4, -2], [2, -9, -3, -1, 21, -1], [7, 3, 3, -6, -8, -2], [-6, 10, -4, -1, 5, -2], [15, -6, -6, -4, 0, 10], [12, -3, -8, 0, -8, -8], [-18, 2, 0, 0, 14, -4], [4, 0, 0, -2, 16, -4], [6, -3, -5, 4, 0, -9], [14, 4, 8, 0, -22, 6], [-4, -8, 2, -3, 3, 8], [6, 4, 8, -7, -13, 10], [7, -1, 5, -7, -15, -8], [22, 10, 4, -10, -8, 2], [28, -7, 3, -2, -2, 9], [-18, 3, 7, -5, -3, -3], [32, -10, -4, 1, 5, -4], [0, -10, 2, -7, -15, -6], [2, -11, -2, -5, 19, -2], [-4, -16, -4, 1, 17, -6], [0, -9, -7, -4, 9, -13], [8, 4, 0, 0, 6, -14], [-32, 2, -6, 2, 4, -14], [-9, -3, -7, 3, 7, -17], [6, -19, 3, 0, 0, 1], [9, 15, -3, 12, 0, 4], [-8, 10, 0, -10, -2, -2], [10, -2, -14, 8, -6, -2], [22, 13, -2, 1, 5, 2], [0, 3, -2, 1, 21, -18], [6, 1, 1, 1, -13, 5], [-9, -5, 11, -3, -1, -5], [-6, 8, 0, 10, 20, -2], [-3, 8, 0, 1, -5, -13], [23, -14, 8, 0, 12, 9], [-12, -13, -1, 7, -11, -3], [7, -2, -1, 8, -2, -2], [18, 8, 2, 6, -12, -2], [2, 5, 9, 3, -11, 11], [0, 16, -6, -4, 14, -6], [18, 13, 9, -3, -19, 1], [6, 2, -1, 5, -15, 5], [4, -4, -6, 0, 22, 2], [-28, 9, -3, -1, -13, -11], [-1, -1, -5, -7, -1, -1], [-16, 8, 0, 2, 0, 0], [-2, 4, -6, -1, -15, -2], [16, -4, 6, -1, -15, 0], [12, -21, 1, 6, 0, 11], [41, -12, 2, -3, 5, 6], [-13, 3, -5, -8, -10, -2], [12, 4, 2, -6, 16, 2], [-8, -12, 4, -10, -2, -4], [4, 8, 6, -5, 13, -10], [-20, 4, -5, -6, -6, 1], [22, -25, 6, -7, 21, 10], [-24, 20, 4, 4, 0, 0], [-25, 8, -4, 0, -12, -4], [2, -7, 3, -4, -14, 7], [-10, -8, -6, -6, 10, -10], [10, 5, -7, 0, -10, -1], [6, -2, 4, -4, -8, -6], [-39, 9, -7, 0, -18, 6], [-12, -9, -5, 4, -4, 5], [-15, -5, 13, -13, -1, 6], [0, -8, 6, 2, 0, -12], [-22, 8, 7, 1, -11, 1], [18, -4, 10, -4, 8, 4], [6, -4, 1, -1, 7, -9], [-5, 4, 9, -9, 3, 4], [-43, 22, -3, 0, -20, 4], [2, 1, 1, 4, 6, 5], [-30, 23, 5, 6, 4, 3], [-12, 11, -3, -2, 7, -1], [6, -2, 4, -2, -24, 6], [14, 2, 10, -4, 4, -6], [-6, 17, 5, 2, -14, 11], [12, 2, 0, 12, -4, -2], [6, -8, 1, 13, -7, -5], [-14, 5, -3, 1, 20, -13], [-10, -11, -5, -5, -5, 3], [4, -4, -4, -6, -10, 0], [-32, 0, 8, 0, -6, -10], [-9, -12, 8, 0, -12, 0], [-40, -4, -6, 9, 9, -8], [5, -17, -5, 1, 5, 9], [22, 1, -5, 4, 2, 5], [-17, 9, -1, 5, -3, -10], [-13, -11, 3, -4, -26, 11], [-38, -8, -4, 2, 8, 6], [27, 2, 10, -2, 2, 2], [29, 2, 9, 6, -4, -6], [-2, 9, 5, -4, 3, -1], [23, -19, 4, -1, 7, 5], [9, -9, 3, 7, -9, 9], [-19, 21, 3, 5, -7, -11], [-12, -1, -11, 8, 16, -5], [16, -4, -8, -2, 2, -8], [-41, 2, 7, -2, -18, 4], [-34, -12, -6, -7, 15, -8], [2, -3, 1, -9, -7, -13], [44, -5, 5, 2, 4, 15], [-18, 7, 1, 1, -13, 11], [-22, 18, 0, 10, -4, 0], [-15, 13, -11, 2, -24, -10], [-18, -12, 0, 0, 6, -2], [22, -5, -10, -3, 21, 10], [26, 7, -9, 8, 12, -13], [-38, 4, 4, 0, 10, -4], [8, 7, 13, -11, -17, 11], [-4, -4, -2, -7, -5, -12], [-8, -4, -12, 4, 2, -2], [-6, 0, -2, -11, 8, 2], [-25, -2, -2, -3, -5, 7], [-10, 1, -3, -13, -16, -3], [38, 10, -2, -4, -2, 2], [8, -20, -6, 1, 9, -4], [1, 11, -8, -1, -5, -1], [16, 12, 8, 6, -10, -4], [-28, 14, -14, -7, -11, -16], [-25, 15, 1, -6, -14, -2], [-17, 7, -1, 8, -2, -14], [14, -13, -1, 6, 5, 5], [-42, -9, -5, -4, 4, 5], [18, -6, 2, 5, 1, -6], [16, -15, 1, -3, -1, -3], [8, 11, 1, 2, -8, 9], [-2, 0, -4, -8, 2, -4], [14, -13, 2, 7, -9, 2], [-58, -7, 3, -1, -11, 3], [4, -23, 7, 2, 13, 19], [-20, 1, -9, 4, 0, -9], [32, 3, -1, 10, -8, 7], [0, 0, 4, 6, -10, 8], [6, 14, 2, 9, -15, -8], [27, -13, 3, 4, 18, 10], [-16, 4, 8, 6, 2, -8], [-12, 5, -8, 11, -15, -6], [-6, -11, 5, -5, 14, -1], [26, 8, 10, 9, -24, -2], [12, 0, 8, 0, -8, 4], [21, 2, 9, 2, -8, 10], [-35, 3, -1, -7, 17, 1], [-12, 17, -5, -5, -17, -3], [16, -18, 2, -9, 17, 0], [-10, 1, 15, 1, 1, 5], [-23, 1, 17, 2, -12, -2], [-18, -20, 2, -9, 5, 0], [28, -10, 8, 2, 14, -6], [18, -3, -11, -4, 12, 7], [-28, -11, 5, -18, 4, 1], [7, 9, 11, 0, -4, -4], [-51, 18, 4, -1, 7, 0], [1, -9, 1, 8, 10, 13], [-30, -11, -7, -2, 19, -17], [-17, 10, 7, 2, 14, -4], [18, -10, -6, 2, 6, 8], [-23, -9, 5, 0, 2, -3], [25, -14, 9, 1, -7, 20], [30, 16, -4, 5, 3, 4], [16, -10, 10, 7, -3, 6], [-15, -14, 0, -9, 15, -16], [-20, 20, 7, -10, -2, 5], [-6, -4, -4, -1, -19, 6], [-11, -14, -11, 8, -6, 18], [-6, 16, 2, 13, -3, -12], [19, -7, 3, -5, 5, 3], [24, -6, 4, 7, -11, 6], [10, 20, -2, 6, -12, -2], [-20, 18, -2, -5, 7, 8], [12, -27, -5, -4, 32, 3], [21, -14, -7, -5, 1, 18], [9, -13, -3, 5, 3, 5], [-5, 13, 5, -3, 5, 3], [-4, -3, -15, -1, 7, 5], [17, -18, 2, -9, 5, 9], [-2, -6, 0, -9, 17, -2], [-22, -24, 10, -12, -10, 4], [-3, 4, 10, 7, -13, 6], [13, -9, -3, 5, 1, 14], [38, -29, -2, 3, 11, 2], [34, -3, 5, 6, -26, 9], [22, 9, 5, 5, 7, -9], [40, 8, 6, -2, -4, 4], [32, 0, 10, 0, -22, 8], [-31, 2, -6, 2, -18, 2], [21, 13, 10, -1, -29, -3], [25, -1, 11, 11, -1, 17], [-22, -6, -2, 3, 15, 4], [6, -4, 6, -4, -28, 24], [20, -6, 2, -9, -3, 12], [36, -23, -3, -1, -1, -3], [9, -15, -14, 8, 0, 7], [-8, -28, 4, -6, 14, 8], [7, -2, -5, 4, 34, -2], [-28, 6, 0, 16, 8, -12], [28, 0, -3, -10, 10, 11], [30, -10, -12, -3, 13, -4], [12, -14, -6, 6, -5, 10], [22, -5, 2, 5, 9, 2], [12, -12, 6, 15, -11, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2366_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2366_2_a_bh();".
// 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_2366_2_a_bh(:prec:=6)
    chi := MakeCharacter_2366_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_2366_2_a_bh();".
// 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_2366_2_a_bh( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2366_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 2, 21, 12, -10, -2, 1]>,<5,R![-71, 10, 93, -24, -19, 2, 1]>],Snew);
    return Vf;
end function;