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

Newform orbit 17.6.a.b

$q$ -expansion

$q + q^{2} - 18q^{3} - 31q^{4} - 16q^{5} - 18q^{6} + 28q^{7} - 63q^{8} + 81q^{9} - 16q^{10} - 138q^{11} + 558q^{12} + 82q^{13} + 28q^{14} + 288q^{15} + 929q^{16} - 289q^{17} + 81q^{18} - 2260q^{19} + 496q^{20} - 504q^{21} - 138q^{22} - 3424q^{23} + 1134q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{3}$ is a root of

$x + 17$

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	−18	−16	28	−138	82	−289	−2260	−3424	8304	−4580	5932	9990	−12776	−768	−12630	37968	18476	−51272	−10592	−70974	−25944	−63056	7706	99662

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ for the primes $p < 1000$ not dividing 17, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

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)[1]   # 17.6.a.b

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

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