Atlas › Modular forms › Weight 8 › Level 38 › 38.8.a › 38.8.a.a

Newform orbit 38.8.a.a

$q$ -expansion

$q - 8q^{2} + 77q^{3} + 64q^{4} + 440q^{5} - 616q^{6} + 951q^{7} - 512q^{8} + 3742q^{9} - 3520q^{10} - 8398q^{11} + 4928q^{12} - 6223q^{13} - 7608q^{14} + 33880q^{15} + 4096q^{16} + 26211q^{17} - 29936q^{18} - 6859q^{19} + 28160q^{20} + 73227q^{21} + 67184q^{22} - 64213q^{23} - 39424q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{5}$ is a root of

$x - 517$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
19	$+1$

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
$\operatorname{tr} a_p$	−8	77	440	951	−8398	−6223	26211	−6859	−64213	65845	−32708	−436694	−28808	650272	58736	−918703	−787635	3106862	2726001	−1800958	−1436223	3402110	−9454038	−40980	4281646

$\operatorname{tr} a_n$ 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 38, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.28 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, 2, 19, 38]: [1, 3, 12, 37] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 4 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(38, 8)[0]   # 38.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 38:8 1000