Atlas › Modular forms › Weight 10 › Level 11 › 11.10.a › 11.10.a.b

Newform orbit 11.10.a.b

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 16q^{2} + 112q^{3} + 708q^{4} + 1594q^{5} + 10378q^{6} + 8400q^{7} + 40716q^{8} + 74789q^{9} + 2986q^{10} + 73205q^{11} + 110288q^{12} + 47214q^{13} - 299852q^{14} - 559436q^{15} - 454776q^{16} - 547238q^{17} - 822418q^{18} - 162940q^{19} - 1913320q^{20} + 825496q^{21} + 234256q^{22} + 3415892q^{23} + 1435932q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{5} - 128x^{4} - 90422x^{3} + 9377514x^{2} + 1528897329x - 38333773422$

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
$\operatorname{tr} a_p$	16	112	1594	8400	73205	47214	−547238	−162940	3415892	5868414	11730396	7021250	5595418	29161940	33703664	−88905666	13747712	274324430	323117752	9655356	159287274	−668342072	378353820	851774166	−240502490

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.284 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, 10] (certified at ell = 2147483171)
T = [(2, 1), (3, 1)]; h of degree 8 = 1 x 8 (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(11, 10)[1]   # 11.10.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 11:10 1000