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

Newform orbit 34.8.a.c

$q$ -expansion

$q + 8q^{2} - 42q^{3} + 64q^{4} + 72q^{5} - 336q^{6} - 580q^{7} + 512q^{8} - 423q^{9} + 576q^{10} - 3906q^{11} - 2688q^{12} - 14086q^{13} - 4640q^{14} - 3024q^{15} + 4096q^{16} - 4913q^{17} - 3384q^{18} + 7172q^{19} + 4608q^{20} + 24360q^{21} - 31248q^{22} + 45696q^{23} - 21504q^{24} + O(q^{25})$

Coefficient field

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

$x - 30$

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	−42	72	−580	−3906	−14086	−4913	7172	45696	−58680	−18340	210596	339270	141248	735456	−1608798	−1465320	411812	−1232464	1750560	−1844638	−8487928	3897192	4278186	4004462

$\operatorname{tr} a_n$ 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)[2]   # 34.8.a.c

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

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