Atlas › Modular forms › Weight 2 › Level 121 › 121.2.a › 121.2.a.a

Newform orbit 121.2.a.a

$q$ -expansion

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

Coefficient field

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

$x - 5$

Atkin–Lehner signs

$p$	$w_p$
11	— ( $11^2 \mid 121$ )

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 121.a (Cremona's 121a). Its optimal curve 121a1 is

$y^2 + xy + y = x^3 + x^2 - 30x - 76$

of rank 0 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	1	2	0	−1	5	−6	2	−9	−2	−3	5	0	2	9	8	−6	2	12	2	10	−6	−9	−13

$\operatorname{tr} a_n$ 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 121, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.182 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, 11, 121]: [0, 2, 12] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; 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(121, 2)[0]   # 121.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 121:2 1000