Atlas › Modular forms › Weight 2 › Level 162 › 162.2.a › 162.2.a.a

Newform orbit 162.2.a.a

$q$ -expansion

$q - q^{2} + q^{4} - 3q^{5} - 4q^{7} - q^{8} + 3q^{10} - q^{13} + 4q^{14} + q^{16} - 3q^{17} - 4q^{19} - 3q^{20} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13}$ is a root of

$x + 22$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	— ( $3^2 \mid 162$ )

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 162.a (Cremona's 162a). Its optimal curve 162a1 is

$y^2 + xy = x^3 - x^2 - 6x + 8$

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
$\operatorname{tr} a_p$	−1	0	−3	−4	0	−1	−3	−4	0	9	−4	−1	6	8	−12	−6	0	−1	−4	−12	11	−16	−12	−3	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 162, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.206 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, 6, 9, 18, 27, 54, 81, 162]: [0, 1, 1, 3, 2, 5, 4, 11, 10, 29] (certified at ell = 2147483179)
T = [(5, 1), (7, 3), (11, 5), (13, 7)]; 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(162, 2)[0]   # 162.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 162:2 1000