Atlas › Modular forms › Weight 2 › Level 888 › 888.2.a › 888.2.a.a

Newform orbit 888.2.a.a

$q$ -expansion

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13} + 9T_{17} + 11T_{19}$ is a root of

$x - 61$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 888$ )
3	$+1$
37	$-1$

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 888.a (Cremona's 888a). Its optimal curve 888a1 is

$y^2 = x^3 - x^2 - 200x - 1044$

of rank 0 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	−4	−1	−3	−5	7	5	1	−8	6	1	−2	12	−2	−1	−4	14	14	4	−11	10	5	−9	−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 888, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.633 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, 37, 74, 111, 148, 222, 296, 444, 888]: [0, 1, 1, 2, 3, 3, 5, 8, 3, 11, 14, 22, 42, 42, 82, 160] (certified at ell = 2147483629)
T = [(5, 1), (7, 3), (11, 5), (13, 7), (17, 9), (19, 11)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 5 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(888, 2)[0]   # 888.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 888:2 1000