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# Argument selection for T694
Authority: the coordinated archived skill manifest is
`agents/runs/20261109T165350Z-build-skills/manifest.json`. Its local and live
copies both have SHA-256
`886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b`.
This is the builder's justification; independent critique remains required.
The quantity is one object: every complex zero, counted with multiplicity,
of the identity-coset polynomial
r_f(X) = integral from 0 to i infinity of f(tau)(tau-X)^(k-2) d tau.
The Fourier convention is q=exp(2*pi*i*tau), a_1=1, trivial character,
even k>=4, and exact level N. X is not rescaled by sqrt(N).
Jin–Ma–Ono–Soundararajan, equations (1.1)–(1.4) and Theorem 1.1, establish
this convention and the circle |X|=N^(-1/2). Paşol–Popa, sections 2 and 3,
distinguish the identity component from the full coset-valued polynomial.
Every parameter is a discrete_index, with the dependencies stated here.
These are arithmetic families of modular eigenforms; no continuous reference
grid or independently selected coupling is present. Finite cutoffs are
noncanonical computational choices, independent of decimal notation.
* N selects exact conductor and k selects weight. Group A contains N=1 and
every nonempty even-weight space through k=26. This includes the minimal
cusp form, every intervening space, both embeddings of the first quadratic
Hecke field at weight 24, and the next rational space. Root degrees stop at
24. The proposed 8 embeddings supply 146 roots.
* Group B contains k=4 and 2<=N<=30. This first eligible weight has quadratic
full period polynomials. The broader level bound covers prime, composite
and nonsquarefree conductors and nonrational Hecke fields at low root cost.
Every embedded newform is selected, rather than only rational or famous
forms. The proposed 61 embeddings supply 122 roots.
* Group C contains k=6 and 2<=N<=15. Quartic polynomials add a second weight
and more critical periods. The smaller conductor bound balances that
extra degree against breadth; the proposed 27 embeddings supply 108 roots.
* Group D contains k=12 and 2<=N<=10. It compares the first full-level cusp
weight across nontrivial levels and gives ten-root configurations, without
taking the Cartesian product of all weights and all levels. The proposed
25 embeddings supply 250 roots. It includes every Hecke field, not only
the rational subset selected for the planned parity-polynomial companions.
The integer i counts ALL embedded normalized newforms in a selected space,
lexicographically by exact real coefficients (a_2,a_3,...). Comparisons through
the Sturm bound distinguish forms. Algebraic real-root isolation and exact
comparison fix indices; software orbit order does not. Empty spaces have no
rows. The roster records field polynomials, rational isolating intervals,
exact Sturm coefficients, Fricke signs, and the exact match to every overlapping
T683 embedded LMFDB label. All three planned tables retain this indexing.
The root index j=1,...,k-2 orders every root by increasing argument in [0,2*pi),
then by modulus in a tie; repeated roots repeat that position consecutively.
The positive real axis has angle zero. The theorem places every root for one
form on the same circle; no independent radius cutoff is used. Roots
proved by functional equations or level-one period relations are evaluated
from those identities, not recognized from small decimal components.
This selection reviews breadth across weight, conductor geometry and Hecke
fields before increasing one weight's degree. It does not claim a documented
application for each particular form. Period-polynomial root distributions
and critical-value constraints are the common identification purpose, as
studied in Jin–Ma–Ono–Soundararajan. The bounds limit arithmetic and polynomial
root-finding cost, rather than target an entry quota. Stein's dimension
formulas, Propositions 6.1 and 6.4, independently verify the complete spaces.
The generator, roster, enumeration and table completeness note must agree.
References checked:
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