Atlas › Modular forms › Weight 8 › Level 14 › 14.8.a › 14.8.a.b

Newform orbit 14.8.a.b

$q$ -expansion

$q + 8q^{2} - 66q^{3} + 64q^{4} - 400q^{5} - 528q^{6} - 343q^{7} + 512q^{8} + 2169q^{9} - 3200q^{10} + 40q^{11} - 4224q^{12} - 4452q^{13} - 2744q^{14} + 26400q^{15} + 4096q^{16} + 36502q^{17} + 17352q^{18} - 46222q^{19} - 25600q^{20} + 22638q^{21} + 320q^{22} - 105200q^{23} - 33792q^{24} + O(q^{25})$

Coefficient field

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

$x + 466$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
7	$+1$

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	−66	−400	−343	40	−4452	36502	−46222	−105200	−126334	−170964	20954	318486	77744	703716	1603278	−1171894	−2068872	−994268	33280	−2971454	−2376168	−2122358	6920346	4952710

$\operatorname{tr} a_n$ 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 14, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.258 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, 7, 14]: [1, 3, 5, 16] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 4 = 1 x 4 (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(14, 8)[1]   # 14.8.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 14:8 1000