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

Newform orbit 11.10.a.a

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 186q^{3} + 912q^{4} - 1824q^{5} - 10956q^{6} - 7260q^{7} - 20064q^{8} + 14553q^{9} - 86724q^{10} - 43923q^{11} - 71040q^{12} - 93258q^{13} + 324264q^{14} + 540330q^{15} + 22656q^{16} + 18678q^{17} + 1997028q^{18} - 1027356q^{19} - 156576q^{20} - 1287924q^{21} + 1674690q^{23} - 947232q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 186x^{2} - 9771x + 82744$

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
$\operatorname{tr} a_p$	0	−186	−1824	−7260	−43923	−93258	18678	−1027356	1674690	−2693658	−4525302	−8820204	−9771102	18795744	−31155816	47500122	332138370	−49031730	330560082	−57835050	−458816886	−798908748	1239784920	−699523368	−2207436012

$\operatorname{tr} a_n$ 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)[0]   # 11.10.a.a

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

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