I guess I am wrong about the degree 240, as it is only 120. You want $\tilde A_5$ (2-extension) and not $\hat A_5$ (4-extension) as the latter gives quadratic determinant, to quote the terminology of Jehanne, with the former as trivial determinant. The 120 still looks too big. The octic fields for $163$ and $277$ are already in Bachoc and Kwon (3.2) matwbn.icm.edu.pl/ksiazki/aa/aa62/aa6211.pdf for $\tilde A_4$. The Magma Artin representation coding takes little time to compute this as a black box via the L-series, if it is what is correct that we are looking for in that case.

I've never been able to discern philosophically the relation of Booker's smooth sums for modularity versus the alternative approach of testing the functional equation. For the latter, you have the completed L-function as a sum $\sum_n F(nxA) + (sign)\sum_n G(nx/A)$ for any $A$ as a 1-parameter family, though Rubinstein can widen the test functions even more I think. Then you just compute at various $A$ values, for instance $A=1.1, 1.2$ to see if the result is self-consistent. Booker draws graphs, but the statistical content of them is not directly transparent.

Yes, now I remember having someone else tell me the same thing about JRS and it not being totally real. Does not Jehanne carry out for totally real though? Viz. Examples 2-5 (page 353-6) for example has $x^5-17*x^3+30*x^2-4*x-7$ and three others of totally real type. A general purpose tool like SAGE/Magma is wrong for a special case, so the original poster would have the right idea to specialize the code.

Actually, Sands, Jehanne, Roblot cover this in section 3.4 of emis.de/journals/EM/expmath/volumes/12/12.4/Roblot.pdf explaining how to compute everything. They quote a paper of Jehanne for the coefficient calculation dx.doi.org/10.1006/jnth.2001.2656 They verify Stark's conjecture, so I guess everything works.

I do not know, but my impression is that for 3-torsion with the Gross curve the tendency is to be Abelian. I tested the same for $d=-23$ (the Gross curve) and got a cyclic group of order 12. But my comments are not from an expert. Computing with 5-torsion is time-consuming. If you look at twists (preserving $j$-invariant), they have larger Galois group. See below.

There is code in Magma packages to do ModularCurveQuotient which is $X_0(N)$ mod Atkin-Lehners, via Galbraith's thesis. The looking at it seems that you can just change ModularForms(N,2) to ModularForms(Gamma1(N),2) in the function internals and hope to work with no Atkin-Lehners. This gives a canonical embedding to $C^{g-1}$ if so. Why you want this for $g=48$ with $X_1(50)$ as 1035 quadrtics is unclear but it ran in 2 minutes.

Somewhat better? Find a local solution with invertible Jacobian, on a subset of equations maybe, and use a Newton technique to lift it high, so that LLL (p-adic) will find it in the number field. I don't think that $R$ or $C$ will help as well as p-adic for converging. What is the expected field degree, and size of the solutions? The largest I have seen with this is 8-10 variables and field degree 50-100. The work should be for finding the local solution.

To precise more: Sur une application d'un theoreme de M. Hadamard, E. Borel, Bull. Sci. Math. 18 (1894), 22-25 archive.org/stream/s2bulletindessci18fran#page/n27/mode/2up

See also jlms.oxfordjournals.org/cgi/reprint/s2-41/2/193.pdf They choose a,q cleverly to gain an extra constant factor in some cases, and sketch a lower-bounding of the constant from Prachar.

As a grad student I ran across a paper of McCurley that referenced Piltz for GRH. Or maybe it was this one: ams.org/journals/mcom/1987-48-177/S0025-5718-1987-0866095-8/… The reference to Piltz (missing from the above wikipedia advert) is A. Piltz, Uber die Haufigkeit der Primzahlen in arithmetischen Progressionen und uber verwandte Gesetze, A. Neuenhahn, Jena, 1884. flipkart.com/book/ber-die-hufigkeit-der-primzahlen/1113365641 (also on GoogleBooks). This was his dissertation, and he also conjectures that $p_n - p_{n-1} < p^\alpha$ for all $\alpha > 0$.

See also Imin Chen. On Siegel's modular curve of level 5 and the class number one problem. Journal of Number Theory, v. 74, no. 2, 1999, 278--297. linkinghub.elsevier.com/retrieve/pii/S0022314X98923204 Burcu Baran. Normalizers of non-split Cartan subgroups, modular curves and the class number one problem, (submitted). mat.uniroma2.it/~baran/classlast.pdf \bysame. A modular curve of level 9 and the class number one problem, Journal of Number Theory, vol. 129 (2009) 715-728 mat.uniroma2.it/~baran/baranarxiv1.pdf The ideas are all a lot the same.

"Comment on your answer: for Grossencharacters of any field, one can consider the job done now, because for silly reasons they vanish identically if the field isn't totally real or CM." Is your terminology like mine? Grossencharacters for totally imaginary but not CM fields exist, but factor through the norm down to the CM-subfield. With an example, the Deligne conjecture was proved by Blasius for CM fields and Harder for totally imaginary. The paper of Harder and Schappacher discusses this, page 36 and on to 43. dx.doi.org/10.1007/BFb0084583