Atlas › Modular forms › Weight 4 › Level 104 › 104.4.a › 104.4.a.b

Newform orbit 104.4.a.b

$q$ -expansion

$q + 5q^{3} + 19q^{5} - 3q^{7} - 2q^{9} - 2q^{11} - 13q^{13} + 95q^{15} + 77q^{17} - 58q^{19} - 15q^{21} + 76q^{23} + O(q^{25})$

Coefficient field

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

$x - 62$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 104$ )
13	$+1$

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	5	19	−3	−2	−13	77	−58	76	−6	−292	207	240	−317	−375	−692	214	−488	782	−1057	1174	892	704	6	830

$\operatorname{tr} a_n$ 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 104, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.115 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, 8, 13, 26, 52, 104]: [1, 2, 3, 5, 5, 13, 24, 46] (certified at ell = 2147483629)
T = [(3, 1), (5, 3)]; h of degree 9 = 1 x 9 (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(104, 4)[1]   # 104.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 104:4 1000