Atlas › Modular forms › Weight 4 › Level 196 › 196.4.a › 196.4.a.b

Newform orbit 196.4.a.b

$q$ -expansion

$q - 4q^{3} - 6q^{5} - 11q^{9} - 12q^{11} + 82q^{13} + 24q^{15} + 30q^{17} - 68q^{19} + 216q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{5}$ is a root of

$x + 10$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 196$ )
7	— ( $7^2 \mid 196$ )

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	−4	−6	0	−12	82	30	−68	216	246	112	110	246	−172	−192	558	−540	−110	140	−840	550	−208	−516	1398	−1586

$\operatorname{tr} a_n$ 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 196, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.139 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, 2, 4, 7, 14, 28, 49, 98, 196]: [1, 2, 3, 3, 8, 15, 15, 44, 87] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 10 = 1 x 10 (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(196, 4)[1]   # 196.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 196:4 1000