Atlas › Modular forms › Weight 2 › Level 124 › 124.2.a › 124.2.a.a

Newform orbit 124.2.a.a

$q$ -expansion

$q - 2q^{3} - 3q^{5} - q^{7} + q^{9} - 6q^{11} + 2q^{13} + 6q^{15} + 6q^{17} - q^{19} + 2q^{21} - 6q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{5}$ is a root of

$x + 5$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 124$ )
31	$-1$

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 124.a (Cremona's 124a). Its optimal curve 124a1 is

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

of rank 1 with torsion of order 3. 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
$\operatorname{tr} a_p$	0	−2	−3	−1	−6	2	6	−1	−6	0	1	−10	−9	8	0	0	−3	−10	−4	−15	14	8	6	12	−7

$\operatorname{tr} a_n$ 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 124, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.194 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, 4, 31, 62, 124]: [0, 1, 2, 3, 10, 19] (certified at ell = 2147482951)
T = [(3, 1), (5, 1)]; h of degree 2 = 1 x 2 (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(124, 2)[0]   # 124.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 124:2 1000