  A class of stochastic Besov spaces $B^p L^2(\Omega;\dot
H^\alpha(\mathcal{O}))$, $1\le p\le\infty$ and $\alpha\in[-2,2]$, is introduced
to characterize the regularity of the noise in the semilinear stochastic heat
equation \begin{equation*} {\rm d} u -\Delta u {\rm d} t =f(u) {\rm d} t + {\rm
d} W(t) , \end{equation*} under the following conditions for some
$\alpha\in(0,1]$: $$ \Big\| \int_0^te^{-(t-s)A}{\rm d} W(s)
\Big\|_{L^2(\Omega;L^2(\mathcal{O}))} \le C t^{\frac{\alpha}{2}}
\quad\mbox{and}\quad \Big\| \int_0^te^{-(t-s)A}{\rm d} W(s) \Big\|_{B^\infty
L^2(\Omega;\dot H^\alpha(\mathcal{O}))}\le C. $$ The conditions above are shown
to be satisfied by both trace-class noises (with $\alpha=1$) and
one-dimensional space-time white noises (with $\alpha=\frac12$). The latter
would fail to satisfy the conditions with $\alpha=\frac12$ if the stochastic
Besov norm $\|\cdot\|_{B^\infty L^2(\Omega;\dot H^\alpha(\mathcal{O}))}$ is
replaced by the classical Sobolev norm $\|\cdot\|_{L^2(\Omega;\dot
H^\alpha(\mathcal{O}))}$, and this often causes reduction of the convergence
order in the numerical analysis of the semilinear stochastic heat equation. In
this article, the convergence of a modified exponential Euler method, with a
spectral method for spatial discretization, is proved to have order $\alpha$ in
both time and space for possibly nonsmooth initial data in
$L^4(\Omega;\dot{H}^{\beta}(\mathcal{O}))$ with $\beta>-1$, by utilizing the
real interpolation properties of the stochastic Besov spaces and a class of
locally refined stepsizes to resolve the singularity of the solution at $t=0$.
