Atlas › Modular forms › Weight 2 › Level 911 › 911.2.a › 911.2.a.a

Newform orbit 911.2.a.a

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q - 5q^{2} + q^{3} + 3q^{4} - q^{5} + 2q^{6} + 5q^{7} - 6q^{8} - 8q^{9} - q^{10} - q^{11} - 7q^{12} - 26q^{13} - 16q^{14} - 14q^{15} - 5q^{16} - 5q^{17} - 8q^{18} - 24q^{19} - 5q^{20} - 8q^{21} + 3q^{22} + 6q^{23} + 18q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} + 4x^{8} - 14x^{7} - 44x^{6} + 74x^{5} + 134x^{4} - 151x^{3} - 145x^{2} + 90x + 59$

Atkin–Lehner signs

$p$	$w_p$
911	$+1$

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
$\operatorname{tr} a_p$	−5	1	−1	5	−1	−26	−5	−24	6	11	−2	3	−10	−10	−43	−8	−15	−8	−35	−3	−24	−22	−19	−36	2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.292 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 911]: [0, 77] (certified at ell = 2147480791)
T = [(2, 1), (3, 1)]; h of degree 76 = 1 x 76 (phi(ord eps) x dim S_k^new); 8 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(911, 2)[0]   # 911.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 911:2 1000