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

Newform orbit 911.2.a.b

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q + 2q^{2} - 7q^{3} + 8q^{4} - 7q^{5} - 13q^{6} - 15q^{7} + 3q^{8} + 7q^{9} - 18q^{10} - 7q^{11} - 4q^{12} + q^{13} + 7q^{14} + 6q^{15} - 4q^{16} - 19q^{17} + 3q^{18} - 6q^{19} - 3q^{20} - 6q^{21} - 13q^{22} - 12q^{23} - 39q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{14} + 5x^{13} - 17x^{12} - 103x^{11} + 82x^{10} + 717x^{9} - 170x^{8} - 2261x^{7} + 437x^{6} + 3469x^{5} - 1143x^{4} - 2255x^{3} + 1153x^{2} + 260x - 167$

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
$\operatorname{tr} a_p$	2	−7	−7	−15	−7	1	−19	−6	−12	−28	−28	−20	−36	−4	47	13	12	−74	32	6	−55	−16	17	32	−52

$\operatorname{tr} a_n$ 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)[1]   # 911.2.a.b

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

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