Coordinate Law Family

This scan compares the full {I,T_3|Q,T_11|Q} coordinate laws of the Heegner operators on the current canonical mod-3 levels and the two nearest false positives for the Atkin-Lehner law.

level	status	`T_3`	`T_7`	`T_11`	`T_19`	`T_43`	`T_67`	`T_163`	all match `163`?
`163`	`canonical`	`[0, 1, 0]`	`[2, 0, 1]`	`[0, 0, 1]`	`[0, 2, 0]`	`[1, 1, 2]`	`[1, 1, 2]`	`[2, 2, 0]`	`True`
`829`	`canonical`	`[0, 1, 0]`	`[1, 1, 1]`	`[0, 0, 1]`	`[0, 2, 2]`	`[2, 0, 0]`	`[0, 2, 0]`	`[1, 1, 0]`	`False`
`2053`	`canonical`	`[0, 1, 0]`	`[1, 1, 2]`	`[0, 0, 1]`	`[2, 0, 0]`	`[0, 2, 0]`	`[0, 2, 1]`	`[1, 1, 1]`	`False`
`3853`	`T3_partial`	`[0, 1, 0]`	`[2, 0, 0]`	`[0, 0, 1]`	`[0, 2, 2]`	`[0, 2, 1]`	`[0, 2, 1]`	`[1, 1, 0]`	`False`
`4933`	`misaligned`	`[0, 1, 0]`	`[2, 0, 0]`	`[0, 0, 1]`	`[0, 2, 0]`	`[2, 0, 0]`	`[1, 1, 0]`	`[1, 1, 0]`	`False`

Read

Exact 163 coordinate-law matches in this sample: [163].
Sample levels that fail at least one Heegner coordinate law: [829, 2053, 3853, 4933].
This kills the stronger coordinate-law conjecture immediately: 829 is canonical but already fails the full 163 Heegner table.
So the stable canonical data is strictly weaker than an exact coordinate packet for all Heegner operators; only the local T_3/T_11 algebra and weaker shadows such as w_N|Q = I + T_3|Q survive this test.