Hypothetical connection of the theta functions of Dirichlet characters with the real cyclotomic fields

We consider a possible approach to the Lindelöf hypothesis for Dirichlet $L$-functions. It is based on a special form of the functional equation for the corresponding theta functions. To estimate $L_χ(0.5+it)$ we need to solve certain systems of linear equations. The entries to the corresponding matrices are formed by the summands to the series for theta functions and their derivatives. Numerical data suggest that the inverse matrices have a deep structure and allow us to state a number of conjectures. In particular, it seems that for a character modulo $q$ certain entries to the inverse matrices tend to finite limits when the sizes of the matrices run over arithmetical progressions with step $2q$. Moreover, these limits belong to the real cyclotomic field $\mathbb{Q}(\cos(π/q))$ (up to a scaling factor of $\sqrt{q}$).