Let A subset W of $P^3$ defined by $W=\{p(x) \in P^3/\int ^3_1 p(x)=0\}$

Question

Let A subset $W$ of $P^3$ defined by $W=\{p(x) \in P^3/\int ^3_1 p(x)=0\}$ Is W is subspace? Find its basis and Dimension ?

My attempt : clearly $W$ is subspace because $ap(x)+bq(x) \in W$

for basis let $p(x)=a_0+a_1x+a_2x^2+a_3x^3$

then $\int ^3_1p(x)=0 \Rightarrow 2a_0+4a_1+\frac{28}{3}a_2+20a_3=0$ from here i am stuck