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Professor Buzzard raises the a question of whether every normalized eigenform of level $1$ is defined over a quadratic extension of $\mathbb{Q}_2$.

(This is Question 4.3 of http://www2.imperial.ac.uk/~buzzard/maths/research/papers/conjs.pdf)

In contrast, the multiplicity of the valuation of the set of $2$-adic slopes at level $1$ can be arbitrarily large, as can be observed as follows:

Consider the space $S_k:=S^{new}_k(\Gamma_0(2))$ of newforms of weight $k$. Every newform has slope $(k-2)/2$. Thus, by work of Coleman, the number of slopes of valuation $(k-2)/2$ at level $1$ and weight $k + 2^n$ for sufficiently large $n$ will be at least $\mathrm{dim}(S_k)$, which is unbounded as $k$ increases.

EDIT: The point of the last example is that the answer to the original question is "not much", i.e., there can be many forms of the same slope, but all the forms are defined over a small (or even trivial) extension of $\mathbb{Q}_2$.

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Professor Buzzard raises the a question of whether every normalized eigenform of level $1$ is defined over a quadratic extension of $\mathbb{Q}_2$.

(This is Question 4.3 of http://www2.imperial.ac.uk/~buzzard/maths/research/papers/conjs.pdf)

In contrast, the multiplicity of the valuation of the set of $2$-adic slopes at level $1$ can be arbitrarily large, as can be observed as follows:

Consider the space $S_k:=S^{new}_k(\Gamma_0(2))$ of newforms of weight $k$. Every newform has slope $(k-2)/2$. Thus, by work of Coleman, the number of slopes of valuation $(k-2)/2$ at level $1$ and weight $k + 2^n$ for sufficiently large $n$ will be at least $\mathrm{dim}(S_k)$, which is unbounded as $k$ increases.