Pictures of how the functional equations of twists by Ramanujan sums pin down a modular form, and where they fall short.
The semicircles in which a functional equation lifts a point cover the line only for N ≤ 4. Ramanujan-sum twists fill the gaps, except at the cusps of the excluded levels.
The matrix of legal rows, level by level. Where the theorem allows, the rank climbs to Q; at the excluded levels an explicit vector is missed by every row.
From A. R. Booker, M. Lee and N. Zubrilina, Towards a twistless converse theorem.