Atlas › Modular forms › Weight 2 › Level 242 › 242.2.a › 242.2.a.b

Newform orbit 242.2.a.b

$q$ -expansion

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43}$ is a root of

$x + 176$

Atkin–Lehner signs

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

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

Elliptic curve

By modularity this newform is the one attached to the isogeny class 242.b (Cremona's 242a). Its optimal curve 242a1 is

$y^2 + xy = x^3 + 3x + 1$

of rank 1 with torsion of order 1. Its $a_p$ , computed by Sagebrush's point-counting engine, equal the newform's for every prime $p < 1000$ .

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
$\operatorname{tr} a_p$	1	−2	−3	−2	0	−5	−3	−2	6	3	2	−7	−3	−8	6	−3	0	10	−10	12	−14	−2	−18	−9	11

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

Sato–Tate

The normalized $a_p / 2p^{1/2}$ 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 0.28 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, 2, 11, 22, 121, 242]: [0, 1, 2, 5, 12, 35] (certified at ell = 2147482921)
T = [(3, 1), (5, 2), (7, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8), (31, 9), (37, 10), (41, 11), (43, 12)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 4 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, 2)[1]   # 242.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 242:2 1000