back to table · edit · history · where entries came from · files · download
3132 bytes, as of the version from 2026-11-09 14:49 (current). Recorded here, not run.
# Argument selection for T690
The quantity is the primitive, uncompleted arithmetic symmetric-square L-value of a normalized level-one holomorphic cuspidal Hecke eigenform. Every finite Euler factor is included. No absolute value, completed value, adjoint shift or Petersson norm is stored as a second quantity.
* `k` is a discrete index: the even weight of a nonzero level-one cusp space. We include every such weight through 26, namely 12, 16, 18, 20, 22, 24, 26. This is a noncanonical computational cutoff, independent of numeral base. It includes the minimal cusp form, all intervening one-dimensional spaces, the first quadratic coefficient field at weight 24, and the next one-dimensional space. This gives field and weight breadth before increasing the cutoff. The dimension formula and the exact Hecke polynomials in T215 support the roster. We do not assert an application for each individual form.
* `i` is a discrete index of an embedded normalized eigenform. Include every embedding in every selected space, sorted by increasing real a_2: one each except for both weight-24 embeddings. For a definition that extends beyond this finite roster, lexicographic Fourier-coefficient order resolves a possible a_2 tie. There are no ties in this roster, so its identities agree with T323/T325/T326. This is a tie-breaking clarification, not a change to existing indices.
* `s` is intrinsic. Include every integer where both the gamma factors at s and the gamma factors at its reflection 2k-1-s are regular. For even k and completion Gamma_C(s) Gamma_R(s-k+2), these are odd 1 <= s <= k-1 and even k <= s <= 2k-2. There are k critical integers per embedded form, hence 162 entries. They are not a sampling grid in a continuous parameter. All values are included, with their signs, including the large left-side values.
The selection has no continuous exploratory reference component and no Cartesian product of unrelated inputs. The only finite cost choice is the weight bound; the embedding and critical-argument sets are complete within each selected weight. Neither the decimal size nor numerical success of a value determines inclusion. Enumeration is increasing k, i, s. Parameter domains describe the infinite mathematical family; the completeness note records the finite cutoff.
Sources checked: the live T215 definition, dimension formula and T_2 polynomials; the live T323, T325 and T326 indexing conventions; SageMath's ModularForm_abstract.symsquare_lseries source and documentation (gamma shifts [0,1,2-k], reflection 2k-1-s); Balkanova and Frolenkov, The mean value of symmetric square L-functions, Algebra & Number Theory 12 (2018), section 2, equations (2-3) through (2-5), with the analytic argument shifted by k-1 to the arithmetic convention here. The paper uses weight 2k where this table uses k.
Archived instructions: ../20261109T122928Z-build-skills/manifest.json. Both archived local and live skills have SHA-256 886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b. The copies were read and verified identical. This is build rationale, not the independent critique's acceptance sidecar.