Atlas › Modular forms › Weight 4 › Level 68 › 68.4.a › 68.4.a.a

Newform orbit 68.4.a.a

$q$ -expansion

$q - 2q^{3} - 8q^{5} - 12q^{7} - 23q^{9} - 10q^{11} - 38q^{13} + 16q^{15} - 17q^{17} + 4q^{19} + 24q^{21} + 120q^{23} + O(q^{25})$

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	— ( $2^2 \mid 68$ )
17	$+1$

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	−2	−8	−12	−10	−38	−17	4	120	56	164	−236	70	−144	48	−366	−504	−460	−768	72	−734	736	856	906	46

$\operatorname{tr} a_n$ 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 68, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.101 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, 17, 34, 68]: [1, 2, 3, 6, 16, 30] (certified at ell = 2147483489)
T = [(3, 1), (5, 1)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 1 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(68, 4)[0]   # 68.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 68:4 1000