An icosahedral representation with conductor 5203 and determinant χ_{−43}

We consider a two-dimensional irreducible representation attached to the following polynomial (example 6 in [J]).

X^{24} + 24X^{22} + 59952X^{20} − 496386X^{18} + 1882872X^{16} − 3827348X^{14} +4261919X^{12} − 2093369X^{10} + 312688X^{8} + 170324X^{6} − 66944X^{4} −3225X^{2} + 1849.

We denote by L(ρ, s) = ∑ a_{n} the L-function of ρ. We have computed for prime numbers ℓ ∉ {11, 43} 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 π̂_{ℓ} a representative of the conjugacy class in Â_{5} of the Frobenius element corresponding to ℓ.

ℓ  π̂_{ℓ}  a_{ℓ}
1  2
α  2i
ω  −2
αω  −2i
19,29,41,101,181,191,193,211,223,251,263,  { \atop (12)(34)}
271,311,397,431,433,439,467,479,487,557,569,593,  or  0
617,619,643,653,673,691,701,709,757,823,839,863  α{ \atop (12)(34)}
31,59,103,269,307,317,509,541,769,787,809,821,827,877,907  { \atop (123)}  −1
3,37,131,277,313,373,419,449,491,571,577,587,607,677,727,811,829  α{ \atop (123)}  −i
17,47,67,109,167,173,229,239,283,337,659,797,883  ω{ \atop (123)}  1
61,71,89,163,241,463,499,503,521,601,631,641,733,761,953  αω{ \atop (123)}  i
23,139,353,359,367,401,443,563,919,967,977  { \atop (12345)}  \frac{−1+\sqrt{5}}{2}
2,7,113,199,389,409,457,887  α{ \atop (12345)} i \frac{−1+\sqrt{5}}{2}
13,127,379,461,719,857  ω{ \atop (12345)}  \frac{1−\sqrt{5}}{2}
137,179,257,331,383,739,911,937,997  αω{ \atop (12345)} i \frac{1−\sqrt{5}}{2}
79,83,97,107,547,661,853,947,971  { \atop (12354)}  \frac{−1−\sqrt{5}}{2}
73,149,157,227,233,347,349,523,751,983,991  α{ \atop (12354)} i \frac{−1−\sqrt{5}}{2}
53,197,281,293,599,613,683,881,941  ω{ \atop (12354)}  \frac{1+\sqrt{5}}{2}
5,151,421,647,743,773,859,929  αω{ \atop (12354)} i \frac{1+\sqrt{5}}{2}

For ramified primes, we have a_{11} = 0 and a_{43} = i.

Bibliography

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