Atlas › Modular forms › Weight 6 › Level 84 › 84.6.a › 84.6.a.b

Newform orbit 84.6.a.b

$q$ -expansion

$q + 9q^{3} - 34q^{5} - 49q^{7} + 81q^{9} - 332q^{11} - 1026q^{13} - 306q^{15} + 922q^{17} + 452q^{19} - 441q^{21} - 3776q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + T_{11}$ is a root of

$x + 366$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 84$ )
3	$-1$
7	$+1$

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	9	−34	−49	−332	−1026	922	452	−3776	1166	−9792	2390	−7230	4652	24672	1110	46892	−9762	−26252	65440	−5606	−9840	61108	−62958	−37838

$\operatorname{tr} a_n$ 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 84, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.302 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, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84]: [1, 2, 3, 4, 7, 5, 13, 12, 16, 23, 44, 86] (certified at ell = 2147483647)
T = [(5, 1), (11, 1)]; h of degree 6 = 1 x 6 (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(84, 6)[1]   # 84.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 84:6 1000