Atlas › Modular forms › Weight 2 › Level 549 › 549.2.a › 549.2.a.c

Newform orbit 549.2.a.c

$q$ -expansion

$q + q^{2} - q^{4} + 3q^{5} + q^{7} - 3q^{8} + 3q^{10} + 5q^{11} + q^{13} + q^{14} - q^{16} - 4q^{17} - 4q^{19} - 3q^{20} + 5q^{22} + 9q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{5}$ is a root of

$x - 4$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 549$ )
61	$+1$

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 549.c (Cremona's 549c). Its optimal curve 549c1 is

$y^2 + xy = x^3 - x^2 - 18x - 27$

of rank 0 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	0	3	1	5	1	−4	−4	9	6	0	8	−5	−8	−4	−6	−9	−1	−7	8	−11	3	−4	4	−14

$\operatorname{tr} a_n$ 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 549, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.212 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, 3, 9, 61, 183, 549]: [0, 1, 2, 5, 22, 64] (certified at ell = 2147482921)
T = [(2, 1), (5, 1)]; h of degree 25 = 1 x 25 (phi(ord eps) x dim S_k^new); 3 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(549, 2)[2]   # 549.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 549:2 1000