For completeness, we recall the following definition from [6]:

*An n-cochain (n ∈ ℤ*_{+}) of a Lie conformal algebra $\mathcal{A}$ with coefficients in an $\mathcal{A}$-module V is a ℂ-linear map
$$\gamma :{\mathcal{A}}^{\otimes n}\to v[{\lambda}_{1},\cdots ,{\lambda}_{n}],\phantom{\rule{thickmathspace}{0ex}}{a}_{1}\otimes \cdots \otimes {a}_{n}\mapsto {\gamma}_{{\lambda}_{1},\cdots ,{\lambda}_{n}}({a}_{1},\cdots ,{a}_{n})$$
*satisfying the following conditions*:

(i) ${\gamma}_{{\text{\lambda}}_{1},\cdots ,{\text{\lambda}}_{n}}({a}_{1},\cdots ,\partial {a}_{i}\cdots ,{a}_{n})=-{\text{\lambda}}_{i}{\gamma}_{{\text{\lambda}}_{1}},\cdots ,{\text{\lambda}}_{n}({a}_{1},\cdots ,{a}_{n})$
*(conformal antilinearity)*,

(ii) *γ is skew-symmetric with respect to simultaneous permutations of a*_{i}'s and λ_{i}'s (skew-symmetry).

As usual, let $\mathcal{A}$^{⊗0} = ℂ, so that a 0-cochain is an element of *V*. Denote by ${\stackrel{~}{C}}^{n}(\mathcal{A},V)$
the set of all *n*-cochains. The differential *d* of an *n*-cochain *γ* is defined as follows:
$$\begin{array}{l}{(}_{d}({a}_{1},\cdots ,{a}_{n+1}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=}{\displaystyle \sum _{i=1}^{n+1}{\left(-1\right)}^{i+1}{a}_{{i}_{{\text{\lambda}}_{i}}}{\gamma}_{{\text{\lambda}}_{1},\cdots ,{\widehat{\text{\lambda}}}_{i},\cdots ,{\text{\lambda}}_{n+1}}({a}_{1,}\cdots ,{\widehat{a}}_{i},\cdots ,{a}_{n+1})}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+}{\displaystyle \sum _{\underset{i<j}{i,j=1}}^{N+1}{(-1)}^{i+j}}{\gamma}_{{\text{\lambda}}_{i}+{\text{\lambda}}_{j},{\text{\lambda}}_{1,}\cdots {\widehat{\text{\lambda}}}_{j},\cdots ,{\text{\lambda}}_{n+1}}([{a}_{{i}_{{\text{\lambda}}_{i}}}{a}_{j}],{a}_{1},\cdots ,{\widehat{a}}_{i},\cdots ,{\widehat{a}}_{j},\cdots ,{a}_{n+1}),\end{array}$$(22)
where *γ* is linearly extended over the polynomials in *λ*_{i}. In particular, if *γ ∈ V* is a 0-cochain, then (dγ)_{λ}(a) = a_{λ}γ.

It is known from [6] that the operator *d* preserves the space of cochains and *d*^{2} = 0. Thus the cochains of a Lie conformal algebra $\mathcal{A}$ with coefficients in its module *V* form a complex, which is denoted by
$${\stackrel{~}{C}}^{\bullet}(\mathcal{A},V)=\underset{n\in {\mathbb{Z}}_{+}}{\oplus}{\stackrel{~}{C}}^{n}(\mathcal{A},V)\text{,}$$(23)
and called the *basic complex*. Moreover, define a (left) ℂ[∂]-module structure on ${\stackrel{~}{C}}^{\bullet}(\mathcal{A},V)$
by
$${(\partial \gamma )}_{{\text{\lambda}}_{1}}{}_{,\cdots ,{\text{\lambda}}_{n}}({a}_{1},\cdots ,{a}_{n})=({\partial}_{v}+{\displaystyle \sum _{i=1}^{n}{\text{\lambda}}_{i}}){\gamma}_{{\text{\lambda}}_{1},\cdots ,{\text{\lambda}}_{n}}({a}_{1},\cdots ,{a}_{n}),$$
where *∂*_{V} denotes the action of *∂* on *V*. Then *d∂ = ∂d* and thus $\mathrm{\partial}{\stackrel{~}{C}}^{\bullet}(\mathcal{A},V)\subset {\stackrel{~}{C}}^{\bullet}(\mathcal{A},V)$
forms a subcomplex. The quotient complex
$${C}^{\bullet}(\mathcal{A},V)={\stackrel{~}{C}}^{\bullet}(\mathcal{A},V)/\mathrm{\partial}{\stackrel{~}{C}}^{\bullet}(\mathcal{A},V)=\underset{n\in {\mathbb{Z}}_{+}}{\oplus}{c}^{n}(\mathcal{A},V)$$
is called the *reduced complex*.

*The basic cohomology*
${\tilde{\text{H}}}^{\u2022}(\mathcal{A},V)$
*of a Lie conformal algebra $\mathcal{A}$ with coefficients in an $\mathcal{A}$-module V is the cohomology of the basic complex* ${\tilde{\text{C}}}^{\u2022}(\mathcal{A},V)$
*and the (reduced) cohomology* ${\text{H}}^{\u2022}(\mathcal{A},V)$
*is the cohomology of the reduced complex* ${\text{C}}^{\u2022}(\mathcal{A},V)$
.

For a *q*-cochain
$\gamma \text{\hspace{0.17em}}\in \text{\hspace{0.17em}}{\tilde{C}}^{q}(\mathcal{A},V)$
we call γ a *q-cocycle* if *d(γ)* = 0; a *q-coboundary* if there exists a (*q* — 1)-cochain
$\varphi \in {\tilde{C}}^{q-1}(\mathcal{A},V)$
such that *γ = d(ϕ)*.
Two cochains γ_{1} and *γ*_{2} are called *equivalent* if γ_{1} — γ_{2} is a coboundary. Denote by ${\tilde{D}}^{q}(\mathcal{A},V)$
and ${\tilde{B}}^{q}(\mathcal{A},V)$
the spaces of *q*-cocycles and *q*-boundaries, respectively. By Definition 5.2,
$${\tilde{\text{H}}}^{q}(\mathcal{A},V)={\tilde{D}}^{q}(\mathcal{A},V)/{\tilde{B}}^{q}(\mathcal{A},V)=(\text{equivalent\hspace{0.17em}classes\hspace{0.17em}of\hspace{0.17em}}q-\text{cocycles})\text{.}$$
The main results of this section are the following.

*For the Lie conformal algebra $\mathcal{W}$, the following statements hold*.

(i) *For the trivial module* ℂ,
$$\mathrm{dim}{\tilde{\text{H}}}^{q}(\mathcal{W},\u2102)=\{\begin{array}{l}1\text{\hspace{0.17em}}if\text{\hspace{0.17em}}q=0,4,5,6\\ 2\text{\hspace{0.17em}}if\text{\hspace{0.17em}}q=3\text{\hspace{0.17em}}\\ 0\text{\hspace{0.17em}otherwise}\end{array}$$(24)
*and*
$$\mathrm{dim}{\text{H}}^{q}(\mathcal{W},\u2102)=\{\begin{array}{l}1\text{\hspace{0.17em}}if\text{\hspace{0.17em}}q=0,6\\ \begin{array}{l}2\text{\hspace{0.17em}}if\text{\hspace{0.17em}}q=2,4,5\text{\hspace{0.17em}}\\ \text{3\hspace{0.17em}}if\text{\hspace{0.17em}}q=3\end{array}\\ 0\text{\hspace{0.17em}otherwise}\text{.}\end{array}$$(25)

(ii) *If a ≠ 0, then* dim H^{q} ($\mathcal{W}$, ℂ_{a}) = 0, *for q* ≥ 0.

(iii) *If a ≠ 0, then* dim H^{q} ($\mathcal{W}$, *M*_{Δ},α) = 0, *for q* ≥ 0.

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