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

Newform orbit 110.6.a.f

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 8q^{2} - 3q^{3} + 32q^{4} - 50q^{5} - 12q^{6} + 237q^{7} + 128q^{8} + 179q^{9} - 200q^{10} - 242q^{11} - 48q^{12} + 298q^{13} + 948q^{14} + 75q^{15} + 512q^{16} + 1433q^{17} + 716q^{18} + 4767q^{19} - 800q^{20} + 2947q^{21} - 968q^{22} + 902q^{23} - 192q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{1321})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{7}$ , whose minimal polynomial is

$x^{2} - 234x + 1800$

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
$\operatorname{tr} a_p$	8	−3	−50	237	−242	298	1433	4767	902	1029	−2095	5251	1416	11162	−22426	−16233	−47662	32955	−7204	−36061	−642	17730	88396	−92297	−247546

$\operatorname{tr} a_n$ 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)[5]   # 110.6.a.f

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

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