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

Newform orbit 17.8.a.a

$q$ -expansion

$q - 2q^{2} + 18q^{3} - 124q^{4} - 10q^{5} - 36q^{6} - 902q^{7} + 504q^{8} - 1863q^{9} + 20q^{10} - 8634q^{11} - 2232q^{12} + 10858q^{13} + 1804q^{14} - 180q^{15} + 14864q^{16} + 4913q^{17} + 3726q^{18} - 784q^{19} + 1240q^{20} - 16236q^{21} + 17268q^{22} + 77330q^{23} + 9072q^{24} + O(q^{25})$

Coefficient field

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

$x - 16$

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
$\operatorname{tr} a_p$	−2	18	−10	−902	−8634	10858	4913	−784	77330	−18210	−237002	230878	−304182	−525032	802752	152862	−1602408	−2601610	1074604	−502298	3648258	−2892174	728104	7931846	−6551038

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

Sato–Tate

The normalized $a_p / 2p^{7/2}$ 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 0.257 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)[0]   # 17.8.a.a

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

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