Atlas › Modular forms › Weight 2 › Level 856 › 856.2.a › 856.2.a.b

Newform orbit 856.2.a.b

$q$ -expansion

$q + q^{3} - 4q^{5} + 2q^{7} - 2q^{9} + 5q^{11} - 5q^{13} - 4q^{15} - 6q^{17} + q^{19} + 2q^{21} - 3q^{23} + O(q^{25})$

Coefficient field

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

$x + 69$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 856$ )
107	$+1$

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 856.b (Cremona's 856b). Its optimal curve 856b1 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	1	−4	2	5	−5	−6	1	−3	−6	−8	3	−5	0	8	−13	−6	−11	10	10	8	13	−12	1	18

$\operatorname{tr} a_n$ 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 856, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.53 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, 107, 214, 428, 856]: [0, 1, 2, 3, 10, 29, 57, 112] (certified at ell = 2147482621)
T = [(3, 1), (5, 3), (13, 5), (17, 7), (19, 9)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 7 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(856, 2)[1]   # 856.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 856:2 1000