Atlas › Modular forms › Weight 2 › Level 416 › 416.2.a › 416.2.a.a

Newform orbit 416.2.a.a

$q$ -expansion

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 3T_{5}$ is a root of

$x - 2$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 416$ )
13	$-1$

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 416.a (Cremona's 416b). Its optimal curve 416b1 is

$y^2 = x^3 - x^2 + 4$

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	−3	−2	1	−3	−2	−4	2	−4	5	−12	−7	9	4	−6	−4	10	15	−2	8	4	2	10

$\operatorname{tr} a_n$ 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 416, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.233 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, 8, 13, 16, 26, 32, 52, 104, 208, 416]: [0, 1, 2, 3, 1, 4, 5, 6, 10, 18, 32, 60] (certified at ell = 2147483497)
T = [(3, 1), (5, 3)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 2 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(416, 2)[0]   # 416.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 416:2 1000