Atlas › Modular forms › Weight 2 › Level 396 › 396.2.a › 396.2.a.b

Newform orbit 396.2.a.b

$q$ -expansion

$q - 2q^{5} + 2q^{7} + q^{11} + 6q^{13} + 4q^{17} - 2q^{19} + 8q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 3T_{7}$ is a root of

$x - 4$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 396$ )
3	— ( $3^2 \mid 396$ )
11	$-1$

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 396.b (Cremona's 396a). Its optimal curve 396a1 is

$y^2 = x^3 - 696x - 8215$

of rank 0 with torsion of order 2. 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	0	−2	2	1	6	4	−2	8	0	0	−6	0	10	0	−14	12	−14	4	0	6	2	−16	14	−2

$\operatorname{tr} a_n$ 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 396, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.26 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, 3, 4, 6, 9, 11, 12, 18, 22, 33, 36, 44, 66, 99, 132, 198, 396]: [0, 1, 1, 2, 3, 2, 2, 5, 5, 5, 6, 9, 9, 16, 14, 30, 40, 78] (certified at ell = 2147482951)
T = [(5, 1), (7, 3)]; h of degree 3 = 1 x 3 (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(396, 2)[1]   # 396.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 396:2 1000