A138551 Moment sequence of t^3 coefficient in det(tI-A) for random matrix A in USp(6).

1, 0, 2, 0, 23, 0, 684, 0, 34760, 0, 2493096, 0, 228253267, 0, 25091028820, 0, 3179942075960, 0, 451649016238160, 0, 70421753109861592, 0, 11869050034269797984, 0, 2136758627313217104448, 0

Offset: 0

Author

Andrew V. Sutherland, Mar 24 2008

Keywords

• nonn

Comments

Let the random variable X be the coefficient of t^3 in the characteristic polynomial det(tI-A) of a random matrix in USp(6) (6x6 complex matrices that are unitary and symplectic). Then a(n) = E[X^n].

Let L_p(T) be the L-polynomial (numerator of the zeta function) of a genus 3 curve C. Under a generalized Sato-Tate conjecture, for almost all C, a(n) is the n-th moment of the coefficient of t^3 in L_p(t/sqrt(p)), as p varies.

Examples

			a(4) = 23 because E[X^4] = 23 for X the t^3 coeff of det(tI-A) in USp(6).
		

Links

• Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.

Crossrefs

Cf. A138540, A138549.

Formula

See Prop. 12 of Kedlaya-Sutherland.