For convenience we state examples of the three main conjectures for moments in families. There are no open questions in this article (except how to prove these conjectures!). For simplicity, we state the conjectures only for integral moments.
Let be a real, primitive, quadratic character to the modulus (i.e. a Kronecker symbol)
Let be a normalized newform of weight and level (where is prime) (we write and let be the associated -function
(with critical strip .) Let
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