Atlas › Modular forms › Weight 2 › Level 338 › 338.2.a › 338.2.a.d

Newform orbit 338.2.a.d

$q$ -expansion

$q + q^{2} - q^{3} + q^{4} - 3q^{5} - q^{6} - 3q^{7} + q^{8} - 2q^{9} - 3q^{10} - q^{12} - 3q^{14} + 3q^{15} + q^{16} - 3q^{17} - 2q^{18} - 6q^{19} - 3q^{20} + 3q^{21} + 6q^{23} - q^{24} + O(q^{25})$

Coefficient field

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

$x + 10$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
13	— ( $13^2 \mid 338$ )

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 338.d (Cremona's 338e). Its optimal curve 338e1 is

$y^2 + xy + y = x^3 + x^2 + 3x - 5$

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
$\operatorname{tr} a_p$	1	−1	−3	−3	0	0	−3	−6	6	0	0	−3	0	1	−3	−6	6	−8	−12	15	−6	10	6	6	−12

$\operatorname{tr} a_n$ 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 338, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.288 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, 13, 26, 169, 338]: [0, 1, 1, 5, 15, 47] (certified at ell = 2147483053)
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(338, 2)[3]   # 338.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 338:2 1000