  For a prime $p$ larger than $7$, the Eisenstein series of weight $p-1$ has
some remarkable congruence properties modulo $p$. Those imply, for example,
that the $j$-invariants of its zeros (which are known to be real algebraic
numbers in the interval $[0,1728]$), are at most quadratic over the field with
$p$ elements and are congruent modulo $p$ to the zeros of a certain truncated
hypergeometric series. In this paper we introduce "theta modular forms" of
weight $k \geq 4$ for the full modular group as the modular forms for which the
first dim$(M_k)$ Fourier coefficients are identical to certain theta series. We
consider these theta modular forms for both the Jacobi theta series and the
theta series of the hexagonal lattice. We show that the $j$-invariant of the
zeros of the theta modular forms for the Jacobi theta series are modulo $p$ all
in the ground field with $p$ elements. For the theta modular form of the
hexagonal lattice we show that its zeros are at most quadratic over the ground
field with $p$ elements. Furthermore, we show that these zeros in both cases
are congruent to the zeros of certain truncated hypergeometric functions.
