Atlas › Modular forms › Weight 4 › Level 110 › 110.4.a › 110.4.a.i

Newform orbit 110.4.a.i

Trace form

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

$2q - 4q^{2} + 7q^{3} + 8q^{4} - 10q^{5} - 14q^{6} + 27q^{7} - 16q^{8} + 59q^{9} + 20q^{10} - 22q^{11} + 28q^{12} + 42q^{13} - 54q^{14} - 35q^{15} + 32q^{16} + 49q^{17} - 118q^{18} + 133q^{19} - 40q^{20} + 183q^{21} + 44q^{22} + 162q^{23} - 56q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{177})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{7} + 5T_{13} + 7T_{17}$ , whose minimal polynomial is

$x^{2} - 641x - 423014$

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
$\operatorname{tr} a_p$	−4	7	−10	27	−22	42	49	133	162	−407	351	−9	40	−514	98	−53	−986	−25	172	−427	646	−706	−492	−273	3222

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.154 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, 7, 4, 11, 20, 58] (certified at ell = 2147482921)
T = [(3, 1), (7, 3), (13, 5), (17, 7)]; h of degree 10 = 1 x 10 (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, 4)[8]   # 110.4.a.i

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

cargo run --release -p sagebrush-web --example atlas -- 110:4 1000