Atlas › Modular forms › Weight 6 › Level 104 › 104.6.a › 104.6.a.a

Newform orbit 104.6.a.a

$q$ -expansion

$q - 23q^{3} - 9q^{5} + 165q^{7} + 286q^{9} + 222q^{11} - 169q^{13} + 207q^{15} - 1123q^{17} - 978q^{19} - 3795q^{21} + 1492q^{23} + O(q^{25})$

Coefficient field

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

$x + 32$

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	−23	−9	165	222	−169	−1123	−978	1492	−6726	−868	−6461	15672	−16889	217	−11932	−3354	−45744	16470	−52345	−22482	−5044	39792	117814	38822

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.285 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, 4, 7, 7, 19, 38, 74] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 4 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, 6)[0]   # 104.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 104:6 1000