Atlas › Modular forms › Weight 6 › Level 26 › 26.6.a › 26.6.a.a

Newform orbit 26.6.a.a

$q$ -expansion

$q - 4q^{2} + 16q^{4} - 14q^{5} - 170q^{7} - 64q^{8} - 243q^{9} + 56q^{10} - 250q^{11} - 169q^{13} + 680q^{14} + 256q^{16} + 1062q^{17} + 972q^{18} - 78q^{19} - 224q^{20} + 1000q^{22} + 1576q^{23} + O(q^{25})$

Coefficient field

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

$x + 42$

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
$\operatorname{tr} a_p$	−4	0	−14	−170	−250	−169	1062	−78	1576	2578	−8654	10986	1050	−5900	−5962	29046	−13922	−32882	−69566	−50542	−46750	−19348	−87438	94170	182786

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ 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 0.223 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, 2, 7, 19] (certified at ell = 2147483629)
T = [(3, 1), (5, 3)]; h of degree 5 = 1 x 5 (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(26, 6)[0]   # 26.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 26:6 1000