I am currently reading the book of Erich Hecke "Lectures on the Theory of Algebraic Numbers", and it is not clear for me to understand his notation of theta series associated to quadratic form over totally real field.
The first question appears in Page 205, formula (175) (I use the English version GTM 77, for Germany version see Page 230), he defined $\theta(t,z;\alpha)=\sum_{u\in \alpha}\exp(-\pi t_p(u^{(p)}+z_p)^2)$. Here, I undersatnd $t_p$ is the variables and $u^{(p)}$ is the embedding of $u$, but I dont really get what he means for $z_p$. He write $z_p=\sum_{q=1}^n\alpha_q^{(p)}u_q$, but I dont know what is $u_q$.
Next question relates to the Transformation formula, in this book Theorem 159, Page 206. Although the author proved the formula when $t_p$ is real, he also claimed this equality holds when $t_p$ is complex with postive real part. Now I want to ask, in the case when $t_p$ is complex with postive real part, is there any assumpution for $z_p$?