Atkin-Lehner Coordinate Law Scan

This scan tests the candidate identity

w_N|Q = I + T_3|Q

on every currently available qdim=3 level, whenever {I,T_3|Q,T_11|Q} is independent.

level	status	E minpoly	N minpoly	W minpoly	W coords in `{I,E,N}`	matches `I+E`?
`163`	`canonical`	`x^2 + 2*x`	`x^2`	`x^2 + 2`	`[1, 1, 0]`	`True`
`811`	`misaligned`	`x^3 + 2*x^2`	`x^3 + x^2 + x`	`x^2 + 2`	`[0, 2, 1]`	`False`
`829`	`canonical`	`x^2 + 2*x`	`x^2`	`x^2 + 2`	`[1, 1, 0]`	`True`
`1093`	`misaligned`	`x^3 + 2*x^2 + x + 1`	`x^3 + 2x^2 + 2x + 2`	`x + 2`	`[1, 0, 0]`	`False`
`1949`	`misaligned`	`x^3 + 2`	`x^3`	`x + 2`	`[1, 0, 0]`	`False`
`1999`	`misaligned`	`x^3 + x^2 + x`	`x^2`	`x^2 + 2`	`[1, 1, 1]`	`False`
`2053`	`canonical`	`x^2 + 2*x`	`x^2`	`x^2 + 2`	`[1, 1, 0]`	`True`
`2341`	`misaligned`	`x^3 + 2*x^2 + x + 2`	`x^3 + x`	`x^2 + 2`	`[1, 1, 1]`	`False`
`2351`	`misaligned`	`x^2 + 2`	`x^3 + 2*x`	`x + 1`	`[2, 0, 0]`	`False`
`2693`	`misaligned`	`x^3 + 2*x^2 + 1`	`x^3 + 2x^2 + 2x + 2`	`x + 2`	`[1, 0, 0]`	`False`
`2719`	`misaligned`	`x^3 + x^2 + 1`	`x^3 + 2x^2 + 2x`	`x^2 + 2`	`[1, 1, 2]`	`False`
`2953`	`misaligned`	`x^3 + 2*x^2`	`x^2 + 2*x`	`x^2 + 2`	`[2, 0, 2]`	`False`
`2971`	`misaligned`	`x^3 + 2*x^2 + x + 2`	`x^3 + x`	`x^2 + 2`	`[1, 1, 1]`	`False`
`3169`	`misaligned`	`x^3 + 2*x^2`	`x^2 + x`	`x^2 + 2`	`[2, 0, 1]`	`False`
`3371`	`misaligned`	`x^2 + 2`	`x^3 + x`	`x + 1`	`[2, 0, 0]`	`False`
`3581`	`misaligned`	`x^2 + 2*x + 1`	`x^3 + 2x^2 + 2x + 1`	`x^2 + 2`	`[2, 2, 1]`	`False`
`3709`	`misaligned`	`x^2 + 2`	`x^3 + x^2 + 2*x`	`x + 1`	`[2, 0, 0]`	`False`
`3767`	`misaligned`	`x^3 + x^2 + x + 2`	`x^3 + 2*x^2 + 1`	`x + 2`	`[1, 0, 0]`	`False`
`3823`	`misaligned`	`x^3 + 2*x + 1`	`x^3 + x^2 + x + 2`	`x + 2`	`[1, 0, 0]`	`False`
`3853`	`T3_partial`	`x^2 + 2*x`	`x^3 + 2*x^2 + x`	`x^2 + 2`	`[1, 1, 0]`	`True`
`4051`	`misaligned`	`x^3 + 2*x^2`	`x^2 + x`	`x^2 + 2`	`[2, 0, 1]`	`False`
`4057`	`misaligned`	`x^3 + x + 1`	`x^3 + x^2 + 2*x`	`x + 1`	`[2, 0, 0]`	`False`
`4099`	`misaligned`	`x^3 + x^2 + 1`	`x^3 + 2x^2 + 2x`	`x^2 + 2`	`[1, 1, 2]`	`False`
`4159`	`misaligned`	`x^3 + x^2 + x`	`x^3 + x^2`	`x + 1`	`[2, 0, 0]`	`False`
`4447`	`misaligned`	`x^2 + 2`	`x^3 + x`	`x^2 + 2`	`[0, 2, 0]`	`False`
`4519`	`misaligned`	`x^3 + 2`	`x^3`	`x + 1`	`[2, 0, 0]`	`False`
`4933`	`misaligned`	`x^2 + 2*x`	`x^2`	`x^2 + 2`	`[1, 1, 0]`	`True`

Counts

T3_partial: 1/1 match w_N|Q = I + E.
canonical: 3/3 match w_N|Q = I + E.
misaligned: 1/23 match w_N|Q = I + E.

Read

Canonical levels tested: [163, 829, 2053].
Canonical levels matching the law: [163, 829, 2053].
Non-canonical levels with the same law: [4933, 3853].
So the law is not sufficient on the enlarged sample.

Degenerate levels are excluded from the main table because the basis {I,E,N} is not independent there, so the coordinate law is not even well-posed on that class.