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

Newform orbit 26.8.a.a

$q$ -expansion

$q - 8q^{2} - 39q^{3} + 64q^{4} + 385q^{5} + 312q^{6} - 293q^{7} - 512q^{8} - 666q^{9} - 3080q^{10} - 5402q^{11} - 2496q^{12} + 2197q^{13} + 2344q^{14} - 15015q^{15} + 4096q^{16} - 21011q^{17} + 5328q^{18} - 27326q^{19} + 24640q^{20} + 11427q^{21} + 43216q^{22} - 63072q^{23} + 19968q^{24} + O(q^{25})$

Coefficient field

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

$x - 346$

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 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	−39	385	−293	−5402	2197	−21011	−27326	−63072	122238	−208396	−442379	58000	−202025	588511	1684336	−442630	−1083608	3443486	2084705	5937890	−6609256	−142740	−6985286	−200762

$\operatorname{tr} a_n$ 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)[0]   # 26.8.a.a

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

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