Atlas › Modular forms › Weight 8 › Level 11 › 11.8.a › 11.8.a.a

Newform orbit 11.8.a.a

Trace form

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

$2q - 8q^{2} - 6q^{3} - 104q^{4} - 470q^{5} - 696q^{6} - 1228q^{7} + 480q^{8} - 36q^{9} + 4280q^{10} + 2662q^{11} + 6072q^{12} + 344q^{13} + 14752q^{14} - 12990q^{15} - 6368q^{16} - 8468q^{17} + 4464q^{18} - 35280q^{19} + 5240q^{20} - 55356q^{21} - 10648q^{22} - 61486q^{23} + 105120q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{15})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} + 14x - 1451$

Atkin–Lehner signs

$p$	$w_p$
11	$-1$

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	−6	−470	−1228	2662	344	−8468	−35280	−61486	179040	−57166	−877698	−283616	275484	1662512	1616484	−2454130	−6019176	−174698	−1151466	885944	3801460	−2282916	−13481970	−68078

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.262 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, 11]: [1, 8] (certified at ell = 2147483171)
T = [(2, 1), (3, 1)]; h of degree 6 = 1 x 6 (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(11, 8)[0]   # 11.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 11:8 1000