Let $p$ be a prime number and $N$ a positive integer not divisible by $p$.

For some easy choices of $p$ and $N$, can anybody provide me with explicit examples of collections $$\{f_k,\quad 2\leq k \leq k_0 \}$$ of q-expansions such that, for each $k$, $f_k$ is the $q$-series of a classical eigenform of weight $k$ and level $Np$, all being members of an ordinary $p$-adic Hida family of tame level $N$? Here $k_0$ is some reasonable upper bound for the weight, and I leave to the reader the meaning of the term "reasonable". Even if $k_0=4$ I will already be happy, and it is also fine if $k$ only runs among some of the integers between $2$ and $k_0$. But of course, the cardinal of $\{ f_k\}$ should be greater than $1$!

I am interested in Hida families which are neither Eisenstein not CM, as explicit examples of those are already well-known.

The simplest example seems to arise when one takes $N=1$ and $p=11$. In this case the associated Hida Hecke algebra is $\mathbf{T}=\Lambda:=\mathbb{Z}_p[[T]]$. I guess that people like Kevin Buzzard, Robert Pollack, William Stein and many others have computed the $q$-expansions $f_k$ in this case, up to some reasonable $k_0$, and I would be happy to have access to these computations.

Yet another interesting example: reading the paper "How can we construct abelian Galois extensions of basic number fields?" by Barry Mazur, he reports on an example computed by Citro and Stein, where $N=1$ and $p=691$. They compute explicitly $\{f_2,f_{12}\}$ where $f_{12}$ is the $p$-stabilization of the discriminant modular form $\Delta$ and $f_2$ turns out to be the only weight 2 newform of level $p$ and character $\omega^{-10}$, where $\omega$ is the Teichmuller character. I would like to know the $q$-expansions $f_k$ in this case for some other integers in between $2$ and $12$. (For integers $k$ larger than $12$ one can possibly compute $f_k$ by multiplying $f_{12}$ by a suitable Eisenstein series of weight $k-12$ and applying Hida's ordinary projector.)

And of course, any other collection of examples is welcome!