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We prove for primes $p\ge 5$ a conjecture of Coleman on the analyticcontinuation of the family of modular functions$\frac{E^\ast_\kappa}{V(E^\ast_\kappa)}$ derived from the family of Eisensteinseries $E^\ast_\kappa$. The precise, quantitative formulation of the conjectureinvolved a certain on $p$ depending constant. We show by an example that theconjecture with the constant that Coleman conjectured cannot hold in generalfor all primes. On the other hand, the constant that we give is also shown notto be optimal in all cases. The conjecture is motivated by its connection tocertain central statements in works by Buzzard and Kilford, and by Roe,concerning the "halo" conjecture for the primes $2$ and $3$, respectively. Weshow how our results generalize those statements and comment on possible futuredevelopments.

A Conjecture of Coleman on the Eisenstein Family