Atlas › Modular forms › Weight 6 › Level 49 › 49.6.a › 49.6.a.a

Newform orbit 49.6.a.a

$q$ -expansion

$q - 10q^{2} + 14q^{3} + 68q^{4} + 56q^{5} - 140q^{6} - 360q^{8} - 47q^{9} - 560q^{10} + 232q^{11} + 952q^{12} + 140q^{13} + 784q^{15} + 1424q^{16} + 1722q^{17} + 470q^{18} + 98q^{19} + 3808q^{20} - 2320q^{22} + 1824q^{23} - 5040q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{3}$ is a root of

$x - 32$

Atkin–Lehner signs

$p$	$w_p$
7	— ( $7^2 \mid 49$ )

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
$\operatorname{tr} a_p$	−10	14	56	0	232	140	1722	98	1824	3418	7644	−10398	17962	10880	−9324	2262	2730	−25648	−48404	−58560	−68082	31784	20538	50582	58506

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ for the primes $p < 1000$ not dividing 49, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.23 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 engine's checks

dim M^+ at levels [1, 7, 49]: [1, 5, 25] (certified at ell = 2147483647)
T = [(2, 1), (3, 3)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 4 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(49, 6)[0]   # 49.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 49:6 1000