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

Newform orbit 26.8.a.e

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 16q^{2} + 87q^{3} + 128q^{4} + 215q^{5} + 696q^{6} + 705q^{7} + 1024q^{8} + 563q^{9} + 1720q^{10} + 614q^{11} + 5568q^{12} + 4394q^{13} + 5640q^{14} + 15115q^{15} + 8192q^{16} + 6623q^{17} + 4504q^{18} + 5538q^{19} + 13760q^{20} - 25805q^{21} + 4912q^{22} - 40536q^{23} + 44544q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{2305})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} - 302x + 2056$

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
$\operatorname{tr} a_p$	16	87	215	705	614	4394	6623	5538	−40536	−296268	−269900	−356733	209690	856353	1779581	1616894	−1926542	−606882	−3098090	−220205	8592556	−1544888	−3576720	−3340108	23438808

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

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)[4]   # 26.8.a.e

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

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