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# Sources, normalization and implementation evidence for T695
Read issue 309 first with `python3 agents/queue.py show 309`.
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Both archived skill copies and the snapshot canonical file have SHA-256
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The defining Eichler integral and the rationality of its parity parts agree
between Jin-Ma-Ono-Soundararajan (2016), equations (1.1)-(1.4), and
Pasol-Popa (2013), section 2 and Proposition 5.11. The latter uses completion
(2*pi)^(-s) Gamma(s) L(f,s), while the former inserts N^(s/2).
The coefficient formula in this table uses arithmetic uncompleted L-values,
so the conductor factor is absent. PARI writes (X-tau)^(k-2), which agrees
with (tau-X)^(k-2) because k is even.
Expanding the binomial gives
c_(w-j) = binomial(w,j) i^(1-j) j! L(f,j+1)/(2*pi)^(j+1), w=k-2.
Even powers correspond to even j and hence odd critical integers. Exact
Sage modular symbols evaluate z^j {0,infinity}; the +1 sign space retains
the even j moments on this path. Binomial weighting and reversing the moment
index produce the stated polynomial, without further sign changes for even j.
The published Delta example gives primitive polynomial
36 X^10 - 691 X^8 + 2073 X^6 - 2073 X^4 + 691 X^2 - 36.
It controls the coefficient order, parity convention and normalization.
For normalized Fricke eigenvalue w_N, the reciprocal identity is
P(X) = -w_N N^(w/2) X^w P(1/(N X)). This is derived coefficientwise from
the completed functional equation and is checked exactly for every row.
The PMC HTML line immediately after (1.4) displays exponent (k-2)/2 on
(sqrt(N) z), inconsistent with polynomial degree. This table uses the
coefficient derivation, not that displayed exponent. The defining integral
and examples agree, so this typographical issue leaves no convention open.
At level one the three-term Manin relation is checked exactly for every row.
Sage constructs exact sign-plus cuspidal new modular symbols. A simple
one-dimensional rational factor supplies a rational period mapping with the
same kernel as the analytic period map. It is unique up to a common scalar.
Clearing denominators, dividing by the coefficient gcd and fixing the leading
sign therefore gives the exact primitive shape, without rational reconstruction
of decimals. Computation uses named imports; a Laurent polynomial-ring import
is needed by Sage's number-field matrix dispatch when the all-embedding roster
reaches weight 24. No `sage.all` import or floating division is used.
Independence: PARI reconstructs all selected forms and compares exact Fourier
coefficients through the Sturm bound (100 coefficients for each rational
form). Its period-polynomial calculation at two working precisions checks
all normalized coefficients. A separate split Mellin implementation evaluates
integer incomplete gamma functions as finite exponential sums using PARI's
Fourier coefficients and an exact Fricke sign. These analytic checks do not
create the exact values and are not represented as certified error bounds.
Delta's integer eta-product coefficients are an independent input control.
The full/new dimensions are also checked from the genus, cusp and elliptic
point formulas of Stein, independently of the decompositions.
Companions were read live: T80, T146, T215, T323, T325, T326, T683, T684,
T690, T691 and T694. T146 concerns Gaussian periods and is unrelated.
T694 is an offered draft, not published. The even table links to its existing
address and explains the missing relative period scalar. The odd companion
is not yet built, so no T-number or link is invented. T683 has no weight-twelve
rows to map; its data is a related critical-value family, not the source of
this table's values. The existing modular form, number theory and polynomial
tags are used. A future `period polynomial` tag could group T694, T695 and the
planned odd-period table; it is not created for one table.
Research URLs and locators:
https://pmc.ncbi.nlm.nih.gov/articles/PMC4790983/ (1.1)-(1.4), Example 1
https://arxiv.org/html/1202.5802 sections 2, 3 and 5.3, Proposition 5.11
https://wstein.org/books/modform/modform/dimension_formulas.html Propositions 6.1, 6.4
https://doc.sagemath.org/html/en/reference/modsym/sage/modular/modsym/space.html#sage.modular.modsym.space.ModularSymbolsSpace.rational_period_mapping
https://doc.sagemath.org/html/en/reference/modsym/sage/modular/modsym/ambient.html#sage.modular.modsym.ambient.ModularSymbolsAmbient.modular_symbol
https://pari.math.u-bordeaux.fr/dochtml/html-stable/Modular_forms.html mfperiodpol, mfeigenbasis, mfatkininit