Analytical evaluation of infinite series I am trying to calculate the infinite series
$$\sum _{n=-\infty }^{\infty } \frac{(-1)^{n+1} e^{-(n-1)^2\pi}}{1-e^{  (2 n-1)\pi}}\simeq -0.0903244354808$$
Are there any any analytical methods to compute such series exactly?
Context: Such series appear as values of certain modular forms.
 A: I'll post this as an answer since it might be a bit too long for a comment.
Given a quadratic imaginary field $K$ there exists a number $\Omega_K \in \mathbb{C^{\times}}$ such that for any (classical) modular form $f$ of weight $k$, and any $\tau \in K$ with positive imaginary part (i.e. your CM points) we have that
$$ f(\tau) \in \bar{\mathbb{Q}} \cdot\Omega_k^{2k}.$$
This is discussed at length in Don Zagiers excellent notes 'Elliptic modular forms and their applications' where the statement above occurs as Proposition 26.
Determining the number $\Omega_K$ given above for the full modular group can be done using the so-called Chowla-Selberg formula (also discussed in the notes by Don Zagier, allthough a complete proof is only sketched). For instance the Chowla-Selberg formula gives
$$E_4(i) = \frac{3\Gamma(\frac{1}{4})^8}{(2\pi)^6}.$$
In your case we are looking at a mock modular form, hence the statement above does not apply directly. That being said, a bit of googling using the frase 'Chowla Selberg mock modular' leads me to the article 'Hecke structures of weakly holomorphic modular forms and their algebraic properties' by
Dohoon Choia and Subong Limb (Journal of number theory, volume 184, p. 428-450). I am not sure it directly relates to your question, but it does seem to derive a certain Chowla-Selberg formula for mock modular forms.
To tie this to your specific series, note that in the context of mock modular forms it appears to be a special case of an Appell-Lerch sum, and these were studied extensively by Sanders Zwegers.
