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# Argument selection for T683
This build follows numberdb-data family #303, read before computation, and the
coordinated skill archived at
`agents/runs/20261102T183459Z-build-skills/manifest.json`. Both the local and
live versions have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
This is the builder's justification, not an independent critique acceptance.
## Quantity and normalization
Each entry is the uncompleted arithmetic value L(f,s), with a1=1, at an
integer 1 <= s <= k-1. The conductor is the exact newform level N>1.
No oldforms, central derivatives, Petersson norms, or completed values are
additional quantities in the entries. The completed function used to check
the functional equation is N^(s/2)(2*pi)^(-s)Gamma(s)L(f,s). Every embedding,
including both members of conjugate pairs, is retained. Exact zeros are
included when the exact Fricke eigenvalue forces them.
## The form parameter and its underlying choices
The public symbolic identity N.k.a.x.n.i follows the LMFDB label convention:
N is level, k weight, a the character orbit, x the form orbit, n the Conrey
character index and i its coefficient-field embedding index. These labels
are not positions in PARI lists. Exact field polynomials, isolating rational
rectangles and Sturm-bound Fourier coefficients are frozen in roster.json.
An exact field isomorphism checks 100 catalogue coefficients for every orbit.
The generator rechecks the Sturm-bound identity and uniqueness of every root
inside its rectangle. Rounded catalogue root locations only choose an exact
algebraic root; they never become high-precision numerical input coefficients.
There are three disjoint arithmetic slices, not a product of all small levels,
weights and characters:
* A: weight 4, trivial nebentypus, exact levels 2 through 30. Include every
orbit and embedding in each newspace, 45 orbits and 61 embeddings. This is
the first even weight beyond elliptic curves and samples prime, composite,
squarefree and nonsquarefree levels, with nonrational coefficient fields
through degree five. The role is discrete_index.
* B: weight 6, trivial nebentypus, exact levels 2 through 15. Include all
20 orbits and 27 embeddings. The second even motivic weight changes the
functional-equation parity and supplies five critical integers per form.
The shorter level extent balances that extra work. The role is
discrete_index. The level bounds of A and B are practical, noncanonical
computation choices, not a claim that the boundary forms are exceptional.
* C: weight 3, primitive odd quadratic character chi_D=(D/n), with negative
fundamental D and |D|<=40, at exact level |D|. The allowed discriminants
are -3,-4,-7,-8,-11,-15,-19,-20,-23,-24,-31,-35,-39,-40. Empty newspaces
produce no rows. Include all 25 orbits and 50 embeddings in the nonempty
spaces. This arithmetic_structure selection supplies odd weight, nontrivial
characters, CM and non-CM forms, and coefficient degrees through eight.
The conductor equality and primitivity link the level and character; they
are not independently crossed with all small weights. The bound 40 is
noncanonical and computational.
All 57 spaces, including empty spaces, were checked with exact PARI newspace
dimensions. Their dimensions total 61,27,50 and their orbit counts 45,20,25,
agreeing with the independently computed LMFDB catalogue. This breadth is
chosen before extending any one slice to large level. No restriction to
rational coefficient fields or famous individual newforms is imposed.
These are low-conductor arithmetic period families with a recognition purpose:
Mellin integrals and critical modular periods are expressed in this arithmetic
normalization. They are not justified merely by library evaluability. Weight
2 is deferred because its elliptic-curve and period conventions already have
separate corpus coverage; general weights and characters remain outside this
finite extent. This build preserves the family's roster for the later,
separate Petersson-norm proposal. No continuous reference sample is present.
## The s parameter
The critical integers 1 through k-1 are intrinsic arguments of a holomorphic
weight-k cuspidal L-function. They are the complete critical set, not a
numerical sampling grid. Every selected embedded form has all of them,
including forced central zeros: 183 values in A,135 in B,100 in C,418 total.
The parameter declaration states the full domain 1<=s<=k-1; the finite form
coverage belongs in the completeness note.
## Sources actually checked
* LMFDB, Labels for classical modular forms, paragraphs defining
N.k.a.x and N.k.a.x.n.i:
https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/Labels
* LMFDB, Reliability and Completeness of classical modular forms:
https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/Reliability
https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/Completeness
These support the catalogue scope, not the new 100-digit computations.
* Belabas and Cohen, Modular Forms in Pari/GP, sections 2,6 and 9.2:
https://arxiv.org/html/1810.00547
* PARI/GP manual, Modular forms, mfinit (newspace flag 0), mfeigenbasis,
mfsturm, mfembed and lfunmf; and the L-functions manual's discussion of
numerical accuracy:
https://pari.math.u-bordeaux.fr/dochtml/html/Modular_forms.html
https://pari.math.u-bordeaux.fr/dochtml/html/_L_minusfunctions.html
* LMFDB exact mf_newforms and mf_hecke_nf records and coefficient-generator
roots from mf_hecke_cc, archived and hashed in source-provenance.json.
## Exhaustive disjoint embedded-label groups
The lists identify all 138 form identities to help an independent critic
assess the mixed selection. That critic must create its own sidecar and
fingerprint all current dependencies; this document grants no approval.
### A
5.4.a.a.1.1
6.4.a.a.1.1
7.4.a.a.1.1
8.4.a.a.1.1
9.4.a.a.1.1
10.4.a.a.1.1
11.4.a.a.1.1
11.4.a.a.1.2
12.4.a.a.1.1
13.4.a.a.1.1
13.4.a.b.1.1
13.4.a.b.1.2
14.4.a.a.1.1
14.4.a.b.1.1
15.4.a.a.1.1
15.4.a.b.1.1
16.4.a.a.1.1
17.4.a.a.1.1
17.4.a.b.1.1
17.4.a.b.1.2
17.4.a.b.1.3
18.4.a.a.1.1
19.4.a.a.1.1
19.4.a.b.1.1
19.4.a.b.1.2
19.4.a.b.1.3
20.4.a.a.1.1
21.4.a.a.1.1
21.4.a.b.1.1
21.4.a.c.1.1
21.4.a.c.1.2
22.4.a.a.1.1
22.4.a.b.1.1
22.4.a.c.1.1
23.4.a.a.1.1
23.4.a.b.1.1
23.4.a.b.1.2
23.4.a.b.1.3
23.4.a.b.1.4
24.4.a.a.1.1
25.4.a.a.1.1
25.4.a.b.1.1
25.4.a.c.1.1
26.4.a.a.1.1
26.4.a.b.1.1
26.4.a.c.1.1
27.4.a.a.1.1
27.4.a.b.1.1
27.4.a.c.1.1
27.4.a.c.1.2
28.4.a.a.1.1
28.4.a.b.1.1
29.4.a.a.1.1
29.4.a.a.1.2
29.4.a.b.1.1
29.4.a.b.1.2
29.4.a.b.1.3
29.4.a.b.1.4
29.4.a.b.1.5
30.4.a.a.1.1
30.4.a.b.1.1
### B
3.6.a.a.1.1
4.6.a.a.1.1
5.6.a.a.1.1
6.6.a.a.1.1
7.6.a.a.1.1
7.6.a.b.1.1
7.6.a.b.1.2
8.6.a.a.1.1
9.6.a.a.1.1
10.6.a.a.1.1
10.6.a.b.1.1
10.6.a.c.1.1
11.6.a.a.1.1
11.6.a.b.1.1
11.6.a.b.1.2
11.6.a.b.1.3
13.6.a.a.1.1
13.6.a.a.1.2
13.6.a.b.1.1
13.6.a.b.1.2
13.6.a.b.1.3
14.6.a.a.1.1
14.6.a.b.1.1
15.6.a.a.1.1
15.6.a.b.1.1
15.6.a.c.1.1
15.6.a.c.1.2
### C
7.3.b.a.6.1
8.3.d.a.3.1
11.3.b.a.10.1
15.3.d.a.14.1
15.3.d.b.14.1
19.3.b.a.18.1
19.3.b.b.18.1
19.3.b.b.18.2
20.3.d.a.19.1
20.3.d.b.19.1
20.3.d.c.19.1
20.3.d.c.19.2
23.3.b.a.22.1
23.3.b.a.22.2
23.3.b.a.22.3
24.3.h.a.5.1
24.3.h.b.5.1
24.3.h.c.5.1
24.3.h.c.5.2
24.3.h.c.5.3
24.3.h.c.5.4
31.3.b.a.30.1
31.3.b.a.30.2
31.3.b.b.30.1
31.3.b.b.30.2
31.3.b.b.30.3
35.3.c.a.34.1
35.3.c.b.34.1
35.3.c.c.34.1
35.3.c.c.34.2
35.3.c.c.34.3
35.3.c.c.34.4
39.3.d.a.38.1
39.3.d.a.38.2
39.3.d.b.38.1
39.3.d.b.38.2
39.3.d.c.38.1
39.3.d.c.38.2
39.3.d.c.38.3
39.3.d.c.38.4
40.3.e.a.19.1
40.3.e.b.19.1
40.3.e.c.19.1
40.3.e.c.19.2
40.3.e.c.19.3
40.3.e.c.19.4
40.3.e.c.19.5
40.3.e.c.19.6
40.3.e.c.19.7
40.3.e.c.19.8