Atlas › Modular forms › Weight 10 › Level 26 › 26.10.a › 26.10.a.b

Newform orbit 26.10.a.b

$q$ -expansion

$q - 16q^{2} + 192q^{3} + 256q^{4} - 1310q^{5} - 3072q^{6} - 5810q^{7} - 4096q^{8} + 17181q^{9} + 20960q^{10} - 4498q^{11} + 49152q^{12} - 28561q^{13} + 92960q^{14} - 251520q^{15} + 65536q^{16} - 237498q^{17} - 274896q^{18} - 913014q^{19} - 335360q^{20} - 1115520q^{21} + 71968q^{22} + 201544q^{23} - 786432q^{24} + O(q^{25})$

Coefficient field

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

$x + 1118$

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
$\operatorname{tr} a_p$	−16	192	−1310	−5810	−4498	−28561	−237498	−913014	201544	1276834	4163770	−18442662	−22601670	11726308	59291534	108158694	−14920154	−57003746	22074010	44416250	265794626	476755484	−505315830	890840634	−802776958

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Sato–Tate

The normalized $a_p / 2p^{9/2}$ 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 0.315 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, 11, 33] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 4 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, 10)[1]   # 26.10.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 26:10 1000