Atlas › Modular forms › Weight 6 › Level 11 › 11.6.a › 11.6.a.a

Newform orbit 11.6.a.a

$q$ -expansion

$q - 4q^{2} - 15q^{3} - 16q^{4} - 19q^{5} + 60q^{6} + 10q^{7} + 192q^{8} - 18q^{9} + 76q^{10} - 121q^{11} + 240q^{12} - 1148q^{13} - 40q^{14} + 285q^{15} - 256q^{16} + 686q^{17} + 72q^{18} - 384q^{19} + 304q^{20} - 150q^{21} + 484q^{22} + 3709q^{23} - 2880q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{3}$ is a root of

$x + 19$

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
$\operatorname{tr} a_p$	−4	−15	−19	10	−121	−1148	686	−384	3709	−5424	−6443	12063	−1528	−4026	7168	−29862	−6461	−16980	29999	31023	1924	65138	−102714	17415	66905

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ for the primes $p < 1000$ not dividing 11, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.202 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, 6] (certified at ell = 2147483171)
T = [(2, 1), (3, 1)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 1 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, 6)[0]   # 11.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 11:6 1000