Atlas › Modular forms › Weight 2 › Level 896 › 896.2.a › 896.2.a.d

Newform orbit 896.2.a.d

$q$ -expansion

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43} + 13T_{47} + 14T_{53} + 15T_{59} + 16T_{61}$ is a root of

$x - 298$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 896$ )
7	$-1$

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 896.d (Cremona's 896c). Its optimal curve 896c1 is

$y^2 = x^3 - 5x - 2$

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

$\operatorname{tr} a_n$ 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 896, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.498 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, 32, 56, 64, 112, 128, 224, 448, 896]: [0, 1, 2, 1, 3, 4, 4, 7, 6, 12, 10, 20, 18, 36, 68, 132] (certified at ell = 2147483137)
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), (47, 13), (53, 14), (59, 15), (61, 16)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 9 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(896, 2)[3]   # 896.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 896:2 1000