An icosahedral representation with conductor 8164 and trivial determinant

We consider a two-dimensional irreducible representation attached to the following polynomial

X^{24} − 13770X^{22} + 52469365X^{20} − 55431544302X^{18} +10080960908026X^{16} − 137975364303634X^{14} + 511570114282349X^{12} −577445967946818X^{10} + 133560924783322X^{8} − 1067593607742X^{6} +1354808661X^{4} − 440826X^{2} + 1.

We denote by L(ρ, s) = ∑ a_{n} the L-function of ρ. We have computed for prime numbers ℓ ∉ {2, 13, 157} lower than 1000 the complex numbers a_{ℓ} =Tr(Frob_{ℓ,ρ}). We give these numbers in the table below. For all prime numbers ℓ, we denote by π̃_{ℓ} a representative of the conjugacy class in Ã_{5} of the Frobenius element corresponding to ℓ.

ℓ  π̃_{ℓ}  a_{ℓ}
937  1  2
659  ω  −2

17,23,31,37,79,109,131,181,199,257,271,283,317,

331,337,353,383,397,431,433,443,457,509,521,557,593, { \atop (12)(34)} 0

599,613,647,661,673,691,701,797,811,823,839,929,971,997

53,61,89,149,163,191,193,227,233,307,401,

419,523,587,653,677,743,773,859,887,907,911,977 { \atop (123)} −1

11,19,67,73,83,127,139,179,229,241,269,293,

367,461,499,541,571,739,751,809,827,941,953,967,983 ω{ \atop (123)} 1

3,41,43,101,107,137,223,239,251,313,389,

409,439,487,503,547,643,733,769,877,919,991 { \atop (12345)} \frac{−1+\sqrt{5}}{2}

7,97,151,263,277,449,467,563,569,577,

619,631,641,683,709,757,853,857,863  ω{ \atop (12345)}  \frac{1−\sqrt{5}}{2}
29,47,59,71,103,173,311,347,349,359,379,463,607,719,727,761,881  { \atop (12354)} \frac{−1−\sqrt{5}}{2}
5,113,167,197,211,281,373,421,479,491,601,617,787,821,829,883,947 ω{ \atop (12354)}  \frac{1+\sqrt{5}}{2}

For ramified primes, we have a_{2} = a_{13} = a_{157} = 0.

Bibliography

[J] A. Jehanne Realization over ℚ of the groups Ã_{5} and Â_{5}, preprint.