Atlas › Modular forms › Weight 2 › Level 784 › 784.2.a › 784.2.a.i

Newform orbit 784.2.a.i

$q$ -expansion

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43}$ is a root of

$x + 130$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 784$ )
7	— ( $7^2 \mid 784$ )

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 784.i (Cremona's 784e). Its optimal curve 784e1 is

$y^2 = x^3 - x^2 - 16x - 1392$

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	2	4	0	0	0	2	−2	−8	2	4	−6	2	−8	−4	−10	6	−4	12	0	14	8	6	−10	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 784, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.01 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, 7, 8, 14, 16, 28, 49, 56, 98, 112, 196, 392, 784]: [0, 1, 2, 1, 3, 4, 4, 7, 5, 12, 16, 20, 31, 60, 116] (certified at ell = 2147483353)
T = [(3, 1), (5, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8), (31, 9), (37, 10), (41, 11), (43, 12)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 6 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(784, 2)[8]   # 784.2.a.i

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

cargo run --release -p sagebrush-web --example atlas -- 784:2 1000