Atlas › Modular forms › Weight 2 › Level 384 › 384.2.a › 384.2.a.e

Newform orbit 384.2.a.e

$q$ -expansion

$q + q^{3} - 4q^{5} - 2q^{7} + q^{9} - 4q^{11} + 2q^{13} - 4q^{15} - 2q^{17} - 8q^{19} - 2q^{21} - 4q^{23} + O(q^{25})$

Coefficient field

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

$x + 102$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 384$ )
3	$-1$

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 384.e (Cremona's 384h). Its optimal curve 384h1 is

$y^2 = x^3 + x^2 - 35x + 69$

of rank 1 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	1	−4	−2	−4	2	−2	−8	−4	0	6	−2	6	0	−4	0	4	14	−4	−12	−10	−10	12	−14	10

$\operatorname{tr} a_n$ 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 384, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.336 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, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 384]: [0, 1, 1, 2, 3, 3, 5, 4, 8, 6, 12, 10, 20, 18, 36, 68] (certified at ell = 2147483489)
T = [(5, 1), (7, 2), (13, 3), (17, 4), (19, 5), (11, 6), (23, 7), (29, 8)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 3 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(384, 2)[4]   # 384.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 384:2 1000