Maybe what I'm asking is well-known to the experts, however I was not able to find a suitable reference. Any pointer to the literature will be appreciated.
For the notation and teminology, I refer to Birkenake and Lange's book Complex Abelian Varieties, especially Chapter 3.
Let $X=V/\Lambda$ be a complex torus, where $V$ is a $\mathbb{C}$-vector space of dimension $g$ and $\Lambda \subset V$ is a lattice. By the Appell-Humbert theorem, any line bundle $L$ on $X$ is of the form $L=\mathcal{L}(H, \, \chi)$, where $H$ is a hermitian form on $V$ which is integer-valued on $\Lambda$ and $\chi \colon \Lambda \to \mathbb{C}$ is a semi-character.
Assume now that $H$ is nondegenerate and call $E$ the alternating form $\textrm{Im}\, H$. A direct sum decomposition $$\Lambda = \Lambda_1 \oplus \Lambda_2$$ is called a decomposition for $H$ if $\Lambda_1$ and $\Lambda_2$ are isotropic with respect to $E$. Associated with such a decomposition, we can define a semi-character $$\chi_0 \colon \Lambda \to \mathbb{C}, \quad \chi_0(\lambda) = e^{\pi i E(\lambda_1, \, \lambda_2)},$$ where $\lambda= \lambda_1 + \lambda_2$ with $\lambda_i \in \Lambda_i$. Since $\chi_0(\Lambda) \subseteq \{\pm 1\}$, it follows that the line bundle $$L_0 := \mathcal{L}(H, \, \chi_0)$$ is a symmetric line bundle algebraically equivalent to $L$ (where symmetric means $(-1)_A^* L_0 \simeq L_0)$. More precisely, there is a point $c \in V$ such that $L=t_c^*L_0,$ where $t_c \colon X \to X$ is the translation by $c$. Moreover, $c$ is uniquely determined up to translation by elements in $$\Lambda(L)=\{v \in V \, | \, E(v, \, \Lambda) \subseteq \mathbb{Z}\}.$$
Such a result shows, among other things, that any line ample bundle is translated of a symmetric one. The element $c$ as above is called a characteristic of $L = \mathcal{L}(H, \, \chi)$ with respect to the fixed decomposition of $\Lambda$. In particular, $0$ is a characteristic for $L_0$.
By construction, every line bundle of characteristic $0$ (with respect to some decomposition of $\Lambda$) is symmetric, but the converse is not true.
Question 1. Let $L = \mathcal{L}(H, \, \chi)$ be an ample, symmetric line bundle on $X$. Are there conditions on the semi-character $\chi$ ensuring that $L$ is of characteristic $0$ with respect to some decomposition of $\Lambda$ for $H$? Note that $\chi(\Lambda) \subseteq \{\pm 1\}$ is necessary (being equivalent to the symmetry of $L$) but not sufficient.
My second question is closely related to the previous one:
Question 2. Let $L= \mathcal{L}(H, \, \chi)$ be a symmetric line bundle , and let $M_k$ $(1 \leq k \leq 2^{2g})$ be the $2$-torsion line bundles on $X$ (all of them are obviously symmetric). How many among the $2g$ symmetric divisors $L \otimes M_k$ are of characteristic $0$ with respect of some decomposition of $\Lambda$ for $H$?
