Atlas › Modular forms › Weight 2 › Level 810 › 810.2.a › 810.2.a.a

Newform orbit 810.2.a.a

$q$ -expansion

$q - q^{2} + q^{4} + q^{5} - 4q^{7} - q^{8} - q^{10} + 3q^{11} - 4q^{13} + 4q^{14} + q^{16} + 3q^{17} + 5q^{19} + q^{20} - 3q^{22} + 6q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{7} + 2T_{11} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47} + 13T_{53} + 14T_{59} + 15T_{61} + 16T_{67}$ is a root of

$x - 142$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	— ( $3^2 \mid 810$ )
5	$-1$

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 810.a (Cremona's 810b). Its optimal curve 810b1 is

$y^2 + xy = x^3 - x^2 + 36x + 120$

of rank 0 with torsion of order 3. 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
$\operatorname{tr} a_p$	−1	0	1	−4	3	−4	3	5	6	6	2	−4	−3	11	0	6	−3	−10	5	6	−7	14	12	6	11

$\operatorname{tr} a_n$ 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 810, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.654 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, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 81, 90, 135, 162, 270, 405, 810]: [0, 1, 1, 1, 3, 2, 3, 4, 5, 4, 10, 8, 11, 10, 22, 20, 29, 58, 56, 166] (certified at ell = 2147482801)
T = [(7, 1), (11, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12), (53, 13), (59, 14), (61, 15), (67, 16)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 6 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(810, 2)[0]   # 810.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 810:2 1000