Function: hgmeulerfactor
Section: hypergeometric_motives
C-Name: hgmeulerfactor
Prototype: GGLD&
Help: hgmeulerfactor(H,t,p,{&e}): (H,t) being a hypergeometric motive,
 returns the Euler factor P_p at the prime p; if present, set e to the valuation
 of the conductor at p.
Doc: $(H,t)$ being a hypergeometric motive, returns the inverse of its
 Euler factor at the prime $p$ and the exponent $e$ of the conductor at $p$.
 This is not implemented when $p$ is a wild prime: the function returns $0$ and
 sets $e$ to $-1$. Caveat: contrary to \kbd{lfuneuler}, this function returns
 the \emph{inverse} of the Euler factor, given by a polynomial $P_{p}$ such that
 the Euler factor is $1 / P_{p}(p^{-s})$.
 \bprog
 ? H = hgminit([5]); \\ cyclotomic parameters [5] and [1,1,1,1]
 ? hgmeulerfactor(H, 1/2, 3)
 %2 = 729*x^4 + 135*x^3 + 45*x^2 + 5*x + 1
 ? hgmeulerfactor(H, 1/2, 3, &e)
 %3 = 729*x^4 + 135*x^3 + 45*x^2 + 5*x + 1
 ? e
 %4 = 0
 ? hgmeulerfactor(H, 1/2, 2, &e)
 %5 = -x + 1
 ? e
 %6 = 3
 ? hgmeulerfactor(H, 1/2, 5)
 %7 = 0  \\ 5 is wild
 @eprog

 If the conductor is small, the wild Euler factors can be computed
 from the functional equation: set \kbd{L = lfunhgm(H,t)} (the complexity
 should be roughly proportional to the conductor) then
 the \kbd{lfuneuler} function should give you the correct Euler factors
 at all primes:
 \bprog
 ? L = lfunhgm(H, 1/2);
 time = 790 ms.  \\ fast in this case, only 5 is wild
 ? lfunparams(L)   \\ ... and the conductor 5000 is small
 %8 = [5000, 4, [-1, 0, 0, 1]]
 ? lfuneuler(L, 5)
 %9 = 1 \\ trivial Euler factor

 ? L = lfunhgm(H, 1/64); lfunparams(L)
 time = 20,122 ms. \\ slower: the conductor is larger
 %10 = [525000, 4, [-1, 0, 0, 1]]

 ? L = lfunhgm(H, 1/128); lfunparams(L)
 time = 2min, 16,205 ms. \\ even slower, etc.
 %11 = [3175000, 4, [-1, 0, 0, 1]]
 @eprog
