Atlas › Modular forms › Weight 6 › Level 110 › 110.6.a › 110.6.a.d

Newform orbit 110.6.a.d

$q$ -expansion

$q + 4q^{2} - 2q^{3} + 16q^{4} + 25q^{5} - 8q^{6} - 204q^{7} + 64q^{8} - 239q^{9} + 100q^{10} - 121q^{11} - 32q^{12} - 380q^{13} - 816q^{14} - 50q^{15} + 256q^{16} + 440q^{17} - 956q^{18} - 52q^{19} + 400q^{20} + 408q^{21} - 484q^{22} - 1782q^{23} - 128q^{24} + O(q^{25})$

Coefficient field

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

$x + 206$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
5	$-1$
11	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = +1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = -1$ : each L-function in the orbit has odd analytic rank.

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
$\operatorname{tr} a_p$	4	−2	25	−204	−121	−380	440	−52	−1782	−7142	−7080	4066	4262	14400	−20170	20346	−8748	28050	−20186	−25692	9288	31976	29560	−53882	−94018

$\operatorname{tr} a_n$ 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 110, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.324 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, 5, 10, 11, 22, 55, 110]: [1, 2, 3, 9, 6, 17, 32, 94] (certified at ell = 2147482921)
T = [(3, 1), (7, 1)]; h of degree 14 = 1 x 14 (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(110, 6)[3]   # 110.6.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 110:6 1000