An icosahedral representation with conductor 7047 and determinant χ_{−3.29}

We consider a two-dimensional irreducible representation attached to the following polynomial (example 7 in [J]).

X^{24} − 30X^{22} + 243X^{20} + 1526X^{18} − 40059X^{16} + 140280X^{14} −220630X^{12} + 153126X^{10} + 17046X^{8} − 72420X^{6} + 10368X^{4} +34452X^{2} + 7569.

We denote by L(ρ, s) = ∑ a_{n} the L-function of ρ. We have computed for prime numbers ℓ ∉ {3, 29} 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
757  αω  −2i
17,79,83,103,139,151,163,173,197,199,223,241,281,317,  { \atop (12)(34)}
331,367,379,401,409,433,463,491,509,569,601,607,643,661,  or  0
673,709,719,751,787,797,839,881,883,937,941,953,971  α{ \atop (12)(34)}
11,13,41,89,109,113,191,277,359,389,439,443,457,641,809,991  { \atop (123)}  −1
53,73,107,157,179,227,239,271,421,577,701  α{ \atop (123)}  −i
67,181,293,311,349,467,479,571,613,631,659,677,827,887  ω{ \atop (123)}  1
19,59,193,211,233,383,419,557,619,683,967,997  αω{ \atop (123)}  i
47,137,263,269,283,397,449,487,523  { \atop (12345)}  \frac{−1+\sqrt{5}}{2}
23,43,127,257,761,769,823  α{ \atop (12345)} i \frac{−1+\sqrt{5}}{2}
563,599,653,691,743,911,919  ω{ \atop (12345)}  \frac{1−\sqrt{5}}{2}
61,71,167,229,307,521,541,829,857,977  αω{ \atop (12345)} i \frac{1−\sqrt{5}}{2}
7,101,251,461,499,503,617,773,811,877,947  { \atop (12354)}  \frac{−1−\sqrt{5}}{2}
5,31,37,97,353,431,733,739,821,853,859,907  α{ \atop (12354)} i \frac{−1−\sqrt{5}}{2}
2,131,313,373,547,983  ω{ \atop (12354)}  \frac{1+\sqrt{5}}{2}
149,337,347,587,593,647,727,863,929  αω{ \atop (12354)} i \frac{1+\sqrt{5}}{2}

For ramified primes, we have a_{3} = 0 and a_{29} = i.

Bibliography

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