Atlas › Modular forms › Weight 2 › Level 110 › 110.2.a › 110.2.a.d

Newform orbit 110.2.a.d

Trace form

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

$2q - 2q^{2} - q^{3} + 2q^{4} + 2q^{5} + q^{6} + q^{7} - 2q^{8} + 11q^{9} - 2q^{10} - 2q^{11} - q^{12} + 4q^{13} - q^{14} - q^{15} + 2q^{16} - 3q^{17} - 11q^{18} + 7q^{19} + 2q^{20} - 17q^{21} + 2q^{22} - 6q^{23} + q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{33})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{7}$ , whose minimal polynomial is

$x^{2} - 2x - 32$

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
$\operatorname{tr} a_p$	−2	−1	2	1	−2	4	−3	7	−6	−3	1	13	0	−8	−6	9	6	−5	16	3	−8	−14	6	3	−14

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.093 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 2, 5, 10, 11, 22, 55, 110]: [0, 1, 1, 3, 2, 5, 8, 22] (certified at ell = 2147482921)
T = [(3, 1), (7, 3)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 1 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, 2)[3]   # 110.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 110:2 1000