Atlas › Modular forms › Weight 2 › Level 872 › 872.2.a › 872.2.a.a

Newform orbit 872.2.a.a

$q$ -expansion

$q - 2q^{3} + q^{5} + q^{9} - 3q^{11} + 4q^{13} - 2q^{15} - 2q^{17} - 5q^{19} + q^{23} + O(q^{25})$

Coefficient field

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

$x - 1$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 872$ )
109	$-1$

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 872.a (Cremona's 872a). Its optimal curve 872a1 is

$y^2 = x^3 + x^2 + 16$

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	−2	1	0	−3	4	−2	−5	1	9	0	−8	−12	−10	−9	4	−4	5	−4	0	−9	−8	−6	13	5

$\operatorname{tr} a_n$ 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 872, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.368 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, 8, 109, 218, 436, 872]: [0, 1, 2, 3, 9, 29, 58, 114] (certified at ell = 2147482801)
T = [(3, 1), (5, 3)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 4 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(872, 2)[0]   # 872.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 872:2 1000