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A138551
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Moment sequence of t^3 coefficient in det(tI-A) for random matrix A in USp(6).
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1
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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See Prop. 12 of Kedlaya-Sutherland.
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EXAMPLE
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a(4) = 23 because E[X^4] = 23 for X the t^3 coeff of det(tI-A) in USp(6).
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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