Atlas › Modular forms › Weight 6 › Level 40 › 40.6.a › 40.6.a.a

Newform orbit 40.6.a.a

$q$ -expansion

$q - 18q^{3} - 25q^{5} + 242q^{7} + 81q^{9} + 656q^{11} - 206q^{13} + 450q^{15} + 1690q^{17} - 1364q^{19} - 4356q^{21} + 2198q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{7}$ is a root of

$x - 224$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 40$ )
5	$+1$

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	−18	−25	242	656	−206	1690	−1364	2198	−2218	−1700	−846	−1818	10534	12074	32586	8668	−34670	−47566	948	−63102	46536	−88778	−104934	−36254

$\operatorname{tr} a_n$ 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 40, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.237 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, 5, 8, 10, 20, 40]: [1, 2, 4, 3, 7, 9, 18, 34] (certified at ell = 2147483629)
T = [(3, 1), (7, 1)]; h of degree 5 = 1 x 5 (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(40, 6)[0]   # 40.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 40:6 1000