Let $\Gamma=PSL(2,\mathbb{Z})$ be the modular group with the usual presentation $\Gamma=\langle S,U,T|\ S^2=U^3=1, T=US\rangle$ where $S=\left(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right)$, $U=\left(\begin{smallmatrix}1&-1\\1&0\end{smallmatrix}\right)$ and $T=\left(\begin{smallmatrix}1&1\\1&0\end{smallmatrix}\right)$. Then $\Gamma$ acts on the vector space $V$ of polynomials of degree $\leq k$ for an even positive integer $k$ by $$P|_k\gamma=(cx+d)^kP\left(\frac{ax+b}{cx+d}\right), \qquad \gamma=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\in \Gamma.$$
A 1-cocyle is a map $f:\Gamma \rightarrow V$ such that $f(\gamma_1\gamma_2)=f(\gamma_1)|\gamma_2+f(\gamma_2) \ \forall \gamma_1,\ \gamma_2 \in \Gamma $. Suppose $f:\Gamma \rightarrow V$ is a map such that $f(T)=0$, $f(S)=Q$ with $Q+Q|S=0, \ Q+Q|U+Q|U^2=0$ then can one extend $f$ to a 1-cocycle?