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

Newform orbit 544.2.a.j

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 2q^{3} - 2q^{5} + 2q^{7} + 3q^{9} + 10q^{11} - 6q^{13} + 8q^{15} + 3q^{17} + 4q^{19} + 6q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43}$ , whose minimal polynomial is

$x^{3} + 40x^{2} - 12160x + 342688$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 544$ )
17	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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	2	−2	2	10	−6	3	4	6	−10	10	−10	6	−8	−4	−6	0	−18	−8	30	14	−10	8	10	−10

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

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)[9]   # 544.2.a.j

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

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