Constructing Galois representations with prescribed Iwasawa $\lambda$-invariant

Let $p\geq 5$ be a prime number. We consider the Iwasawa $\lambda$-invariants associated to modular Bloch-Kato Selmer groups, considered over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. Let $g$ be a $p$-ordinary cuspidal newform of weight $2$ and trivial nebentype. We assume that the $\mu$-invariant of $g$ vanishes, and that the image of the residual representation associated to $g$ is suitably large. We show that for any number greater $n$ greater than or equal to the $\lambda$-invariant of $g$, there are infinitely many newforms $f$ that are $p$-congruent to $g$, with $\lambda$-invariant equal to $n$. We also prove quantitative results regarding the levels of such modular forms with prescribed $\lambda$-invariant.