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

Newform orbit 11.8.a.b

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 35q^{3} + 604q^{4} + 537q^{5} + 1558q^{6} + 170q^{7} - 420q^{8} + 1823q^{9} + 1470q^{10} - 5324q^{11} - 34360q^{12} + 4250q^{13} - 29988q^{14} + 6841q^{15} + 52744q^{16} + 54300q^{17} - 71350q^{18} + 67844q^{19} + 15888q^{20} + 102262q^{21} - 9015q^{23} + 140844q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} + 35x^{3} - 6789x^{2} - 49545x + 101574$

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 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
$\operatorname{tr} a_p$	0	−35	537	170	−5324	4250	54300	67844	−9015	234078	189857	127895	289842	704930	−1729080	1098660	−4665777	310610	−3368245	−3416541	11466230	566282	−4220790	18265191	11425325

$\operatorname{tr} a_n$ 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)[1]   # 11.8.a.b

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

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