Atlas › Modular forms › Weight 8 › Level 34 › 34.8.a › 34.8.a.a

Newform orbit 34.8.a.a

$q$ -expansion

$q - 8q^{2} - 68q^{3} + 64q^{4} + 270q^{5} + 544q^{6} + 680q^{7} - 512q^{8} + 2437q^{9} - 2160q^{10} - 3756q^{11} - 4352q^{12} - 3994q^{13} - 5440q^{14} - 18360q^{15} + 4096q^{16} + 4913q^{17} - 19496q^{18} - 6628q^{19} + 17280q^{20} - 46240q^{21} + 30048q^{22} - 70056q^{23} + 34816q^{24} + O(q^{25})$

Coefficient field

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

$x - 202$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
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
$\operatorname{tr} a_p$	−8	−68	270	680	−3756	−3994	4913	−6628	−70056	−74250	−291280	511214	−73830	−534268	−967536	−103362	−2133900	−149098	4655564	−5571000	−1991302	8241632	−216612	6106026	6221858

$\operatorname{tr} a_n$ 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 34, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.277 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, 17, 34]: [1, 3, 12, 34] (certified at ell = 2147483489)
T = [(3, 1), (5, 1)]; h of degree 8 = 1 x 8 (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(34, 8)[0]   # 34.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 34:8 1000