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

Newform orbit 49.4.a.d

$q$ -expansion

$q + 2q^{2} + 7q^{3} - 4q^{4} + 7q^{5} + 14q^{6} - 24q^{8} + 22q^{9} + 14q^{10} - 5q^{11} - 28q^{12} - 14q^{13} + 49q^{15} - 16q^{16} - 21q^{17} + 44q^{18} + 49q^{19} - 28q^{20} - 10q^{22} - 159q^{23} - 168q^{24} + O(q^{25})$

Coefficient field

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

$x - 23$

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
$\operatorname{tr} a_p$	2	7	7	0	−5	−14	−21	49	−159	58	147	219	350	−124	525	303	−105	−413	415	−432	−1113	−103	1092	−329	−882

$\operatorname{tr} a_n$ 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)[3]   # 49.4.a.d

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

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