I have a commutative algebra $A$ over the complex numbers. I know that $\mathbb{C}[t]$ is a subalgebra of $A$, and that for any natural number $n$, the quotient $A/(t-n)$ is finite dimensional (and non zero). I do not know anything about the quotients $A/(t-c)$ where $c$ is any other complex number.

Can I deduce that $A$ is finitely generated as an algebra over $\mathbb{C}$? Is there anything which can be said about the sequence of numbers $a_n=dim_{\mathbb{C}}(A/(t-n))$?

Many thanks!