I'm wondering if there is a way to approximate the first $M$ terms of an $\ell^2$ sequence $\{c_n\}$ if we know $|c|_p^p = \sum c_n^p$ for $p=2,3,4,\dots$?

I've tried truncating the sequence at $c_{M+Q}$ for $Q>1$, and applying Newton's method the the $M+Q$-dimensional system of equations $|c|_p^p = \sum^{M+Q} c_n^p$, $p=2,\dots,M+Q+1$, but the jacobian is ill conditioned and this blows up. Is there a better way? Do you know of any work done on this problem?

Thanks!