Atlas › Modular forms › Weight 2 › Level 616 › 616.2.a › 616.2.a.d

Newform orbit 616.2.a.d

$q$ -expansion

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{13} + 7T_{17} + 9T_{19} + 11T_{23}$ is a root of

$x + 4$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 616$ )
7	$+1$
11	$+1$

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 616.d (Cremona's 616a). Its optimal curve 616a1 is

$y^2 = x^3 + 85x + 86$

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

$\operatorname{tr} a_n$ 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 616, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.392 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, 8, 11, 14, 22, 28, 44, 56, 77, 88, 154, 308, 616]: [0, 1, 2, 1, 3, 2, 4, 5, 7, 9, 12, 10, 16, 28, 54, 104] (certified at ell = 2147482951)
T = [(3, 1), (5, 3), (13, 5), (17, 7), (19, 9), (23, 11)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 4 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(616, 2)[3]   # 616.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 616:2 1000