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

Newform orbit 26.10.a.a

$q$ -expansion

$q - 16q^{2} - 273q^{3} + 256q^{4} + 1015q^{5} + 4368q^{6} + 3955q^{7} - 4096q^{8} + 54846q^{9} - 16240q^{10} - 50998q^{11} - 69888q^{12} - 28561q^{13} - 63280q^{14} - 277095q^{15} + 65536q^{16} + 509757q^{17} - 877536q^{18} - 626574q^{19} + 259840q^{20} - 1079715q^{21} + 815968q^{22} + 653524q^{23} + 1118208q^{24} + O(q^{25})$

Coefficient field

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

$x - 742$

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	−273	1015	3955	−50998	−28561	509757	−626574	653524	−4943006	4071700	2348883	−13350960	−7834847	−39637681	73200924	−141141614	−132061256	−185673110	224452625	−172523674	−643288156	720077280	−73028106	−15879778

$\operatorname{tr} a_n$ 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)[0]   # 26.10.a.a

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

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