Atlas › Modular forms › Weight 2 › Level 984 › 984.2.a › 984.2.a.d

Newform orbit 984.2.a.d

$q$ -expansion

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{13} + 7T_{19} + 9T_{23} + 11T_{29}$ is a root of

$x - 15$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 984$ )
3	$-1$
41	$+1$

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 984.d (Cremona's 984d). Its optimal curve 984d1 is

$y^2 = x^3 + x^2 + 7x + 27$

of rank 1 with torsion of order 1. 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	0	−2	−3	−6	−7	0	2	3	1	5	−1	−3	7	6	−12	−7	10	9	−11	−10	12	−6	−12

$\operatorname{tr} a_n$ 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 984, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.051 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, 24, 41, 82, 123, 164, 246, 328, 492, 984]: [0, 1, 1, 2, 3, 3, 5, 8, 4, 12, 16, 24, 46, 46, 90, 176] (certified at ell = 2147482921)
T = [(5, 1), (7, 3), (13, 5), (19, 7), (23, 9), (29, 11)]; h of degree 20 = 1 x 20 (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(984, 2)[3]   # 984.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 984:2 1000