Let
p
p
be a prime, and let
M
\mathcal {M}
denote the space of weight two modular forms on
Γ
0
(
p
)
\Gamma _{0}(p)
all of whose Fourier coefficients are integral, except possibly for the constant term, which should be either integral or half-integral. We prove that
M
\mathcal {M}
is spanned as a
Z
\mathbb {Z}
-module by theta series attached to the unique quaternion algebra that is ramified at
p
p
, at infinity, and at no other primes.