  In this work, Ramanujan type congruences modulo powers of primes $p \ge 5$
are derived for a general class of products that are modular forms of level
$p$. These products are constructed in terms of Klein forms and subsume
generating functions for $t$-core partitions known to satisfy Ramanujan type
congruences for $p=5,7,11$. The vectors of exponents corresponding to products
that are modular forms for $\Gamma_{1}(p)$ are subsets of bounded polytopes
with explicit parameterizations. This allows for the derivation of a complete
list of products that are modular forms for $\Gamma_{1}(p)$ of weights $1\le k
\le 5$ for primes $5\le p \le 19$ and whose Fourier coefficients satisfy
Ramanujan type congruences for all powers of the primes. For each product
satisfying a congruence, cyclic permutations of the exponents determine
additional products satisfying congruences. Common forms among the exponent
sets lead to products satisfying Ramanujan type congruences for a broad class
of primes, including $p> 19$. Canonical bases for modular forms of level $5\le
p \le 19$ are constructed by summing weight one Hecke Eisensten series of
levels $5\le p \le 19$ and expressing the result as a quotient of Klein forms.
Generating sets for the graded algebras of modular forms for $\Gamma_{1}(p)$
and $\Gamma(p)$ are formulated in terms of permutations of the exponent sets. A
sieving process is described by decomposing the space of modular forms of
weight $1$ for $\Gamma_{1}(p)$ as a direct sum of subspaces of modular forms
for $\Gamma(p)$ of the form $q^{r/p}\Bbb Z[[q]]$. Since the relevant bases
generate the graded algebra of modular forms for these groups, the weight one
decompositions determine series dissections for modular forms of higher weight
that lead to additional classes of congruences.
