Atlas › Modular forms › Weight 2 › Level 840 › 840.2.a › 840.2.a.j

Newform orbit 840.2.a.j

$q$ -expansion

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = 6665T_{11} + 1937T_{13} + 5856T_{17} + 9453T_{19} + 4431T_{23} + 7435T_{29} + 859T_{31} + 3987T_{37}$ is a root of

$x - 56338$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 840$ )
3	$-1$
5	$-1$
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 840.j (Cremona's 840j). Its optimal curve 840j1 is

$y^2 = x^3 + x^2 - 15x + 90$

of rank 0 with torsion of order 4. 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	1	−1	4	−2	2	4	0	−2	8	−2	2	4	0	−10	−12	6	12	0	−6	−8	4	2	−14

$\operatorname{tr} a_n$ 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 840, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 3.547 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, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840]: [0, 1, 1, 2, 1, 3, 1, 3, 3, 5, 4, 4, 6, 4, 8, 7, 10, 6, 10, 12, 12, 18, 16, 22, 20, 32, 30, 40, 56, 56, 108, 208] (certified at ell = 2147483629)
T = [(11, 6665), (13, 1937), (17, 5856), (19, 9453), (23, 4431), (29, 7435), (31, 859), (37, 3987)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 8 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(840, 2)[9]   # 840.2.a.j

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

cargo run --release -p sagebrush-web --example atlas -- 840:2 1000