An icosahedral representation with conductor 7947 and trivial determinant

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

X^{24} − 1719X^{22} + 1033803X^{20} − 247759929X^{18} + 18891275922X^{16} −520186579740X^{14} + 4132529834850X^{12} − 12489586256925X^{10} +17037668338668X^{8} − 10424803822722X^{6} + 2281972718133X^{4} −481611663X^{2} + 729.

We denote by L(ρ, s) = ∑ a_{n} the L-function of ρ. We have computed for prime numbers ℓ ∉ {3, 883} 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}
1  2
ω  −2

5,23,67,73,101,107,127,151,173,179,181,193,229,233,283,

293,307,311,313,353,383,491,523,557,571,593,619,641,647, { \atop (12)(34)} 0

677,683,691,701,719,727,733,739,751,761,853,857,971,983

7,31,43,61,103,109,131,137,211,241,277,281,349,401,  { \atop (123)}  −1

431,443,467,509,587,599,607,631,653,673,797,907,919,997

59,71,79,83,97,197,239,269,271,347,379,389,479,503, ω{ \atop (123)} 1

541,563,643,659,743,757,787,821,823,839,877,881,929,941

2,13,37,149,163,251,331,397,419,569,577,613,773,947,991  { \atop (12345)}  \frac{−1+\sqrt{5}}{2}
41,89,113,139,157,191,223,227,263,439,449,457,461,601,617,661,809,863,967 ω{ \atop (12345)}  \frac{1−\sqrt{5}}{2}
17,47,53,167,337,359,373,409,433,487,709,811,887,911,937,953  { \atop (12354)} \frac{−1−\sqrt{5}}{2}
11,19,29,199,257,317,367,421,463,499,521,547,769,827,829,859,977  ω{ \atop (12354)}  \frac{1+\sqrt{5}}{2}

For ramified primes, we have a_{3} = a_{883} = 0.

Bibliography

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