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

Newform orbit 110.6.a.g

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 12q^{2} - 7q^{3} + 48q^{4} + 75q^{5} + 28q^{6} + 273q^{7} - 192q^{8} + 212q^{9} - 300q^{10} - 363q^{11} - 112q^{12} - 204q^{13} - 1092q^{14} - 175q^{15} + 768q^{16} - 627q^{17} - 848q^{18} - 1725q^{19} + 1200q^{20} + 4171q^{21} + 1452q^{22} - 732q^{23} + 448q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{7}$ , whose minimal polynomial is

$x^{3} - 266x^{2} - 24000x + 7336800$

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 even 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
$\operatorname{tr} a_p$	−12	−7	75	273	−363	−204	−627	−1725	−732	5475	17661	21471	15102	8868	−12144	33915	12498	6465	−71370	14091	1140	130122	−118086	132441	219552

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

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)[6]   # 110.6.a.g

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

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