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Twisted Hecke L-values and period polynomials
Let be an orthogonal basis for the space of cusp forms of even
weight on . Let and denote the
-function of and its twist by a Dirichlet character ,
respectively. In this note, we obtain a ``trace formula'' for the values
at integers and with and
proper parity. In the case N=1 or N=2, the formula gives us a convenient way to
evaluate precisly the value of the ratio for a Hecke
eigenform .Comment: 20 page
Differential equations satisfied by modular forms and K3 surfaces
We study differential equations satisfied by modular forms associated to
, where are genus zero subgroups of
commensurable with , e.g., or
. In some examples, these differential equations are realized as
the Picard--Fuch differential equations of families of K3 surfaces with large
Picard numbers, e.g., . Our method rediscovers some of the
Lian--Yau examples of ``modular relations'' involving power series solutions to
the second and the third order differential equations of Fuchsian type in [14,
15].Comment: Some revisions are incorporated, in particular, replaced the
terminology ''bi-modular'' by ''modular'
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