Atlas › Modular forms › Weight 2 › Level 544 › 544.2.a › 544.2.a.c

Newform orbit 544.2.a.c

$q$ -expansion

$q - 2q^{7} - 3q^{9} - 4q^{11} + 2q^{13} - q^{17} - 4q^{19} - 6q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43}$ is a root of

$x - 70$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 544$ )
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 544.c (Cremona's 544a). Its optimal curve 544a1 is

$y^2 = x^3 - 5x + 4$

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	−2	−4	2	−1	−4	−6	8	2	4	−2	4	−12	−6	−4	4	−4	6	−6	10	12	−10	−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 544, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.713 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, 16, 17, 32, 34, 68, 136, 272, 544]: [0, 1, 2, 3, 4, 2, 6, 6, 12, 22, 40, 76] (certified at ell = 2147483489)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (13, 5), (19, 6), (23, 7), (29, 8), (31, 9), (37, 10), (41, 11), (43, 12)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 6 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(544, 2)[2]   # 544.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 544:2 1000