Atlas › Modular forms › Weight 2 › Level 812 › 812.2.a › 812.2.a.a

Newform orbit 812.2.a.a

$q$ -expansion

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

Coefficient field

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

$x + 231$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 812$ )
7	$-1$
29	$+1$

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 812.a (Cremona's 812b). Its optimal curve 812b1 is

$y^2 = x^3 - x^2 - 36x + 232$

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	−1	1	1	1	2	−4	−6	−1	−5	−4	−4	−11	3	3	2	−2	−4	−10	0	−15	−6	4	4

$\operatorname{tr} a_n$ 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 812, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.016 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, 7, 14, 28, 29, 58, 116, 203, 406, 812]: [0, 1, 2, 1, 4, 7, 3, 9, 18, 22, 64, 126] (certified at ell = 2147483353)
T = [(3, 1), (5, 2), (17, 3), (19, 4), (23, 5), (11, 6), (13, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12)]; h of degree 14 = 1 x 14 (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(812, 2)[0]   # 812.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 812:2 1000