We present new information-theoretic protocols for $s$-server, doubly-efficient private information retrieval (PIR) in sublinear server time and one extra bit of space on the servers, for every constant $s \geq 2$. On an $n$-bit database, each of the $s$ PIR servers first builds and stores a preprocessing data structure consisting of $n+1$ bits (which is unique to each server, and after which the server may discard the original database). Then, each server answers a PIR query in time $T_s(n)$, by reading at most $T_s$ bits of its data structure and sending them to the user. In the two server setting ($s=2$), our PIR protocols achieve $T_2 = n^{0.82}$, matching the server time of the state-of-the-art schemes of Henzinger and Ragavan (EUROCRYPT 2026) while shrinking their storage from $O(n\sqrt{\log n})$ to $n+1$ bits. With $s \geq 3$ servers, our PIR protocols retain the $n+1$ bits of storage and additionally improve on the server time of all prior information-theoretic $s$-server schemes with $n^{1+o(1)}$ storage: for example, achieving $T_3 = n^{0.74}$ and $T_{10} = n^{0.61}$ (previously $n^{0.79}$ and $n^{0.75}$). Asymptotically, as the server count $s$ grows, we shrink the number of servers required to achieve a given server-work exponent from $s^{3.9}$ to $s$. Our results draw inspiration from matching-vector techniques (Efremenko, STOC 2009), which we generalize to handle degree-two computations in the exponent. Using this generalization, we build doubly-efficient PIR from the new intermediate abstraction of Fourier-sparse cone-delta functions. Then, we construct such Fourier-sparse cone-delta functions via a connection to modular forms and complex analysis: if a Fourier-sparse cone-delta function does not exist, then there exists a modular form which has too many roots, contradicting the valence formula.

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