Atlas › Modular forms › Weight 8 › Level 17 › 17.8.a › 17.8.a.b

Newform orbit 17.8.a.b

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + q^{2} - 86q^{3} + 225q^{4} - 198q^{5} - 1524q^{6} - 1558q^{7} - 4419q^{8} + 1099q^{9} - 5146q^{10} + 5542q^{11} - 1808q^{12} - 15050q^{13} - 1728q^{14} + 5988q^{15} - 5423q^{16} + 14739q^{17} + 99117q^{18} - 7480q^{19} + 70262q^{20} + 100868q^{21} + 18676q^{22} - 194838q^{23} + 59544q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 85x^{2} + 1002x + 3196$

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 odd 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
$\operatorname{tr} a_p$	1	−86	−198	−1558	5542	−15050	14739	−7480	−194838	−225486	197310	−859374	769806	1018856	−1430440	788122	1371096	589626	−4851452	6699398	444438	1039946	909784	6024450	−12851130

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.257 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, 12] (certified at ell = 2147483489)
T = [(2, 1), (3, 1)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 3 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, 8)[1]   # 17.8.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 17:8 1000