Atlas › Modular forms › Weight 8 › Level 26 › 26.8.a › 26.8.a.b

Newform orbit 26.8.a.b

$q$ -expansion

$q + 8q^{2} - 87q^{3} + 64q^{4} + 321q^{5} - 696q^{6} - 181q^{7} + 512q^{8} + 5382q^{9} + 2568q^{10} + 7782q^{11} - 5568q^{12} + 2197q^{13} - 1448q^{14} - 27927q^{15} + 4096q^{16} + 9069q^{17} + 43056q^{18} - 37150q^{19} + 20544q^{20} + 15747q^{21} + 62256q^{22} + 19008q^{23} - 44544q^{24} + O(q^{25})$

Coefficient field

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

$x - 234$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
13	$-1$

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	−87	321	−181	7782	2197	9069	−37150	19008	174750	29012	323669	795312	−314137	−447441	−1469232	1627770	−2399608	−64066	−322383	−4454782	753560	−1219092	3390330	1628774

$\operatorname{tr} a_n$ 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 26, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.27 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, 13, 26]: [1, 3, 9, 27] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 7 = 1 x 7 (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(26, 8)[1]   # 26.8.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 26:8 1000