Part of it maybe the 5s and 6s effect. In that random sample, the max is 222, in there are 9 fives and 1 six, giving an "extra" four count of 36+14 above independence. The import of duplicates in the 138180 is similar, as a twofer guarantees an extra 28 non-independent for some lottery occurrence. Triplicates are likely too (80% chance).

Maier-Pomerance indicate that they expect $z(\log z)^2$ as the limit (see 1.5), if one knew prime $k$-tuples. They basically use an on-average version of that (in AP), in the paper improving the constant. Thus for large primes, they can't show that any of them individually sieves out more than 1 number, but on average they can show at least 1.31, and Pintz 2. When knowing prime $k$-tuples, at least with uniformity enough, the large primes would then be shown to be more optimal, in sieving out.

My recollection is that if you take level 122, nontrivial real character, weight 2 newform over $Q(i)$ -- then $L(f,1)=0$ but the sign is merely some random number on the unit circle. It is an example to investigate. I don't know whether $L'(f,1)$ is meaningful. Usually I expect, as per David Loeffler that the 0th derivative is nonvanishing. What does this mean in the Heegner construction context?

See Schappacher's book, Periods of Hecke characters (chapter un on Motives). dx.doi.org/10.1007/BFb0082094 See also the thread mathoverflow.net/questions/33269/fontaine-mazur-for-gl-1

Fermat should do this home.bway.net/lewis And I think Magma can too, via either eliminating variables sequentially, or possibly with EliminationIdeal. They have an online calculator, if the problem is not too bulky.

Magma does too much more and different than you want. The $a_i$ shall come from the global desingularization. Then the HomAdjoints will give canonical sheaf sections. If you work just at one alone singularity, you might need to rip open the Magma code to see how they do it. The main file is package/Geometry/SrfHyp/surface_resolution.m

You might mean a Gelfand model. See Garge and Oesterle, and references degruyter.com/view/j/jgth.2010.13.issue-3/jgt.2009.060/…