It depends on the implementation, but the Kronecker-product matrices should be sparse, so the dimension is not always a worry.

I also think philosophically to blame the Heegner construction on modular forms is incorrect, as it is really only the Gross-Zagier formula (not merely the Shimura theory on CM points) that shows that you truly obtain a non-torsion point via the protocol. Gross-Zagier touches more than just modular forms in their proof. In the other direction, if someone asked "Why is descent important?", would an answer be: it gives us a generic semi-practical method of finding points on elliptic curves? I think Fisher has 12-descent working by now, and has found points of height more than 1000 via it.

Heegner points only work for (analytic) rank 1 to find points, largely over Q, and still take time linear in the conductor. For the "visible" range of Cremona's database (as an initial segment of the infinite), the simple searches and descents are now more practical, even after one applies the many tricks (Atkin-Lehner and more) to the Heegner protocol. I would guess that for about 5% of the curves of Cremona (with points) it were found by Heegner methods. What I think you mean is that the Heegner construction is the method that is //proven// to work, thanks to the formula of Gross and Zagier.

It was 500-750MB in Magma for a random short polynomial choice, taking 100-200s to find the CharacteristicPolynomial. f:=Polynomial([Random([-100..100]) : i in [0..36]]); g:=Polynomial([Random([-100..100]) : i in [0..24]]); g:=Polynomial(Reverse(Coefficients(g))); f:=PolynomialRing(Rationals())!f/LeadingCoefficient(f); g:=PolynomialRing(Rationals())!g/LeadingCoefficient(g); P:=KroneckerProduct(CompanionMatrix(f),CompanionMatrix(g)); time char_poly:=CharacteristicPolynomial(P); But if you have larger coefficients it might explode.

This is not a real reference as it is pop-sci, but: newscientist.com/article/… The cited paper is "Sophisticated Euclidean maps in forest chimpanzees" dx.doi.org/10.1016/j.anbehav.2009.01.025 Animal Behaviour, Volume 77, Issue 5, May 2009, Pages 1195-1201 Emmanuelle Normanda and Christophe Boescha. This was on page 1 of a search with Google "study geometry primates", so I am not an expert. It is more about "spatial orientation" than geometry in a mathematical sense.

Actually, I am now confused by your normalization of the half-line. Are the coefficients bounded by 1, so that the symmetry line is $Re(s)=1/2$, and the edge is $Re(s)=1$. This is normalization that Ogg uses.

"...another approach to the Ramanujan conjecture, essentially through Langlands functoriality. ... if we knew for all $n$ that the $L$-functions $L(sym^n f)$ were holomorphic and nonvanishing in the halfplane $Re(s)\ge 1$, the Ramanujan conjecture [follows]..." I don't know about functorality - a variant of the weaker claim is in a paper of Ogg springerlink.com/content/k24u6q718u3v770w Ogg, A. P. A remark on the Sato-Tate conjecture. Invent. Math. 9 1969/1970 198--200. Hs shows a holomorphic continuation past 1/2-line (follows from functorality?) implies no zeros on 1-line.

You can also de-Magma-ize the code with little difficulty. The only hard work was computing the Euler factors at bad primes via the decomposition, now that it done once. The rest appeals to factoring polynomials over finite fields. The LSeries internals is not too deep, as you merely require the Mellin transforms for $\Gamma(s/2)^2$ and $\Gamma((s+1)/2)^2$, which are some variant of Bessel function I suppose, and the Magma code is nothing special for them but uses Dokchitser's general paper. Naming the special values at negative integers applies by LLL as with algdep in PARI.

I guess you want $113317^{202000^{102007}}$ modulo 622301. Well, do the inner exponentiation modulo $\phi(622301)$...

I think I have it set up in Magma now. What do you get for the first 100 prime coeffs of $\rho_0$? My code is not very fast, maybe about 4000 coefficients per second.

The $c$ of Siegel is ineffective. Instead via $L$-functions you get $2^{hR}= 2^{2L(1\,chi)\sqrt d}$, so you need $(2\log 2)L(1,\chi)\sqrt D\ge\log (D/4)$ if I am correct. But this is not known effectively, as the best bounds are from the Gross-Zagier theorem and Goldfeld's work, and would be $\log(D)^{1-\epsilon}$ on the right.

Hmm, I guess g24 with itself does work, though I don't see why g5 is not better. PARI takes no time with g5 and g24, so it is Magma at 10 seconds that is slow I guess.

"X=polcompositum(g24,g24)" you must mean g5 and g24? I agree that is the way to go, I was ignoring that we knew g5. Magma was also around 10 seconds with CompositeFields, giving one answer.

Even it this works, I don't know if Dokchitser's code will really be too fast in practice to get a billion terms. The splitting field (degree 120) was found in 1.5 hours.

They come from the Laplacian, whereas holomorphic forms derive from Cauchy-Riemann. I don't know how to put it briefly more than that. So you get $-y^2 (\partial_{xx} + \partial_{yy}$ giving the Bessel-sincos functions. In general you have $K_{iv}$ where $v$ is related to the eigenvalue as $\lambda=1/4 + R^2$, but you have $v=0$ (eigenvalue 1/4) and cosine (even) here.