Atlas › Modular forms › Weight 4 › Level 49 › 49.4.a › 49.4.a.b

Newform orbit 49.4.a.b

$q$ -expansion

$q - q^{2} + 2q^{3} - 7q^{4} - 16q^{5} - 2q^{6} + 15q^{8} - 23q^{9} + 16q^{10} - 8q^{11} - 14q^{12} - 28q^{13} - 32q^{15} + 41q^{16} - 54q^{17} + 23q^{18} + 110q^{19} + 112q^{20} + 8q^{22} + 48q^{23} + 30q^{24} + O(q^{25})$

Coefficient field

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

$x - 5$

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
$\operatorname{tr} a_p$	−1	2	−16	0	−8	−28	−54	110	48	−110	−12	−246	−182	128	−324	−162	−810	488	244	−768	702	440	1302	−730	−294

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ 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 0.096 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, 3, 15] (certified at ell = 2147483647)
T = [(2, 1), (3, 3)]; h of degree 8 = 1 x 8 (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(49, 4)[1]   # 49.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 49:4 1000