Atlas › Modular forms › Weight 4 › Level 242 › 242.4.a › 242.4.a.e

Newform orbit 242.4.a.e

$q$ -expansion

$q + 2q^{2} + 4q^{3} + 4q^{4} + 3q^{5} + 8q^{6} + 8q^{7} + 8q^{8} - 11q^{9} + 6q^{10} + 16q^{12} + 83q^{13} + 16q^{14} + 12q^{15} + 16q^{16} + 123q^{17} - 22q^{18} - 112q^{19} + 12q^{20} + 32q^{21} + 36q^{23} + 32q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ is a root of

$x - 393$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
11	— ( $11^2 \mid 242$ )

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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	4	3	8	0	83	123	−112	36	−21	128	107	−201	308	−492	−345	204	470	−760	900	−742	92	−864	−645	299

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 242, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.55 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, 2, 11, 22, 121, 242]: [1, 2, 4, 11, 34, 101] (certified at ell = 2147482921)
T = [(3, 1), (5, 2), (7, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 12 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(242, 4)[4]   # 242.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 242:4 1000