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

Newform orbit 242.4.a.m

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 4q^{2} + 6q^{3} + 8q^{4} + 4q^{5} + 12q^{6} + 42q^{7} + 16q^{8} + 38q^{9} + 8q^{10} + 24q^{12} - 4q^{13} + 84q^{14} - 210q^{15} + 32q^{16} + 90q^{17} + 76q^{18} + 122q^{19} + 16q^{20} + 52q^{21} - 146q^{23} + 48q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{37})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ , whose minimal polynomial is

$x^{2} - 636x - 45729$

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
$\operatorname{tr} a_p$	4	6	4	42	0	−4	90	122	−146	44	−266	−536	−194	316	−410	−168	−420	−404	−206	200	816	222	−1430	1346	−2242

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.55 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)[12]   # 242.4.a.m

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

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