  We study the Mahler measures of certain families of Laurent polynomials in
two and three variables. Each of the known Mahler measure formulas for these
families involves $L$-values of at most one newform and/or at most one
quadratic character. In this paper, we show, either rigorously or numerically,
that the Mahler measures of some polynomials are related to $L$-values of
multiple newforms and quadratic characters simultaneously. The results suggest
that the number of modular $L$-values appearing in the formulas significantly
depends on the shape of the algebraic value of the parameter chosen for each
polynomial. As a consequence, we also obtain new formulas relating special
values of hypergeometric series evaluated at algebraic numbers to special
values of $L$-functions.
