Atlas › Modular forms › Weight 2 › Level 816 › 816.2.a › 816.2.a.i

Newform orbit 816.2.a.i

$q$ -expansion

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = 3697T_{5} + 8016T_{7} + 4295T_{11} + 3553T_{13} + 8636T_{19} + 1949T_{23} + 2396T_{29} + 8529T_{31}$ is a root of

$x + 127604$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 816$ )
3	$-1$
17	$+1$

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 816.i (Cremona's 816i). Its optimal curve 816i1 is

$y^2 = x^3 + x^2 - 1621x + 24623$

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	−4	−3	3	−1	−1	−3	−10	−6	−4	5	1	2	−14	6	8	12	−12	2	14	−6	16	0

$\operatorname{tr} a_n$ 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 816, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.77 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, 6, 8, 12, 16, 17, 24, 34, 48, 51, 68, 102, 136, 204, 272, 408, 816]: [0, 1, 1, 2, 3, 3, 5, 4, 2, 8, 6, 12, 8, 12, 22, 22, 42, 40, 80, 152] (certified at ell = 2147483489)
T = [(5, 3697), (7, 8016), (11, 4295), (13, 3553), (19, 8636), (23, 1949), (29, 2396), (31, 8529)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 10 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(816, 2)[8]   # 816.2.a.i

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

cargo run --release -p sagebrush-web --example atlas -- 816:2 1000