# Sources and convention reconciliation for T694

The skill authority is `agents/runs/20261109T165350Z-build-skills/manifest.json`.
Both valid archived versions have SHA-256
886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b.

Jin–Ma–Ono–Soundararajan (2016), equations (1.1)–(1.4), use the integral
in issue #309. Expanding the binomial gives coefficient
binomial(k-2,m) i^(1-m) m! L(f,m+1)/(2*pi)^(m+1).
The conductor-completed Lambda includes N^(s/2); the ordinary moments do not.
PARI's `(X-tau)^(k-2)` is identical because k-2 is even.
The functional equation was derived by reversing coefficients, giving exponent
k-2 on `(sqrt(N)*X)`. The PMC HTML line following equation (1.4) displays
(k-2)/2 there; that exponent is inconsistent with the degree and with the
coefficient formula. We did not copy it. The derived coefficient identity is
checked for every selected polynomial. The definition itself is unambiguous.

Paşol–Popa (arXiv:1202.5802), section 2, gives the two period relations.
At level one, the order-three matrix sends X to 1-1/X and yields
r(X)+X^w r(1-1/X)+(X-1)^w r(-1/(X-1))=0.
For w divisible by 6, evaluation at either root of X^2-X+1 gives 3r(X)=0.
The reality/reflection identity adds their negatives across the imaginary axis.
This proves all four `(+-1 +- i sqrt(3))/2` roots in weights 20 and 26.
The same relations at X=1, combined with constant/leading coefficient
cancellation, give the roots +-1 in every selected level-one weight.
The reciprocal equation at X=+-i/sqrt(N) gives the forced imaginary roots
when epsilon=-1. These arguments determine exact components before storage.

Stein, Dimension Formulas, Propositions 6.1 and 6.4, were implemented
independently as genus/cusp/elliptic-point formulas and old/new subtraction.
They match all 64 selected spaces including empty spaces, and their new
forms total 121 embeddings and 626 roots. T683's published roster was read
through the website attachments, not a campaign transcript. Its weight-four
and weight-six forms match exactly through the Sturm bound, including an
exact algebraic comparison across field presentations. It supplies labels,
not numerical approximations to the new roots.

Research URLs:
https://pmc.ncbi.nlm.nih.gov/articles/PMC4790983/
https://arxiv.org/html/1202.5802
https://wstein.org/books/modform/modform/dimension_formulas.html
https://pari.math.u-bordeaux.fr/dochtml/html-stable/Modular_forms.html
https://www.lmfdb.org/knowledge/show/cmf.label