Atlas › Modular forms › Weight 6 › Level 17 › 17.6.a › 17.6.a.c

Newform orbit 17.6.a.c

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 3q^{2} + 28q^{3} + 69q^{4} + 80q^{5} + 184q^{6} + 284q^{7} - 171q^{8} + 68q^{9} - 678q^{10} - 412q^{11} - 1184q^{12} + 472q^{13} - 1676q^{14} - 744q^{15} - 959q^{16} + 1156q^{17} + 743q^{18} + 3544q^{19} + 230q^{20} + 3216q^{21} + 136q^{22} + 5180q^{23} + 144q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} - 31x^{3} - 322x^{2} + 12668x - 58616$

Atkin–Lehner signs

$p$	$w_p$
17	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	3	28	80	284	−412	472	1156	3544	5180	−6144	3060	1968	18744	15416	−12032	−88696	1848	8048	1168	149396	−60392	25356	121800	−190888	−11800

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.209 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 17]: [1, 8] (certified at ell = 2147483489)
T = [(2, 1), (3, 1)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 2 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(17, 6)[2]   # 17.6.a.c

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 17:6 1000