Atlas › Modular forms › Weight 6 › Level 72 › 72.6.a › 72.6.a.e

Newform orbit 72.6.a.e

$q$ -expansion

$q + 34q^{5} - 240q^{7} + 124q^{11} + 46q^{13} - 1954q^{17} - 1924q^{19} - 2840q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 3T_{7}$ is a root of

$x + 686$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 72$ )
3	— ( $3^2 \mid 72$ )

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	0	34	−240	124	46	−1954	−1924	−2840	8922	−4648	−4362	2886	11332	−7008	22594	28	−6386	−39076	54872	21034	26632	−56188	−64410	−116158

$\operatorname{tr} a_n$ 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 72, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.152 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, 8, 9, 12, 18, 24, 36, 72]: [1, 2, 3, 4, 7, 7, 6, 13, 17, 24, 33, 64] (certified at ell = 2147483647)
T = [(5, 1), (7, 3)]; 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(72, 6)[4]   # 72.6.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 72:6 1000