Atlas › Modular forms › Weight 2 › Level 996 › 996.2.a › 996.2.a.b

Newform orbit 996.2.a.b

$q$ -expansion

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

Coefficient field

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

$x - 3$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 996$ )
3	$-1$
83	$+1$

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 996.b (Cremona's 996c). Its optimal curve 996c1 is

$y^2 = x^3 + x^2 - 12x + 36$

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
$\operatorname{tr} a_p$	0	1	−3	2	−3	−4	0	−7	−3	−6	8	11	0	−4	−12	−3	−9	−1	−13	6	−10	8	−1	3	8

$\operatorname{tr} a_n$ 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 996, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.389 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, 12, 83, 166, 249, 332, 498, 996]: [0, 1, 1, 2, 3, 5, 8, 23, 30, 45, 88, 174] (certified at ell = 2147483323)
T = [(5, 1), (7, 3)]; h of degree 14 = 1 x 14 (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(996, 2)[1]   # 996.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 996:2 1000