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

Newform orbit 14.8.a.a

$q$ -expansion

$q - 8q^{2} - 82q^{3} + 64q^{4} + 448q^{5} + 656q^{6} - 343q^{7} - 512q^{8} + 4537q^{9} - 3584q^{10} + 2408q^{11} - 5248q^{12} + 7116q^{13} + 2744q^{14} - 36736q^{15} + 4096q^{16} + 2486q^{17} - 36296q^{18} + 36482q^{19} + 28672q^{20} + 28126q^{21} - 19264q^{22} - 12880q^{23} + 41984q^{24} + O(q^{25})$

Coefficient field

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

$x - 366$

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 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	−82	448	−343	2408	7116	2486	36482	−12880	−88094	282636	−214534	−140874	36464	716868	−56946	−2149862	3084360	−3034364	−106624	988930	3415896	−15142	174810	13506790

$\operatorname{tr} a_n$ 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)[0]   # 14.8.a.a

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

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