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

Newform orbit 110.6.a.h

Trace form

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

$3q + 12q^{2} + 25q^{3} + 48q^{4} + 75q^{5} + 100q^{6} + 159q^{7} + 192q^{8} + 212q^{9} + 300q^{10} + 363q^{11} + 400q^{12} + 444q^{13} + 636q^{14} + 625q^{15} + 768q^{16} + 1023q^{17} + 848q^{18} + 789q^{19} + 1200q^{20} + 1449q^{21} + 1452q^{22} + 2064q^{23} + 1600q^{24} + O(q^{25})$

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

$x^{3} - 184x^{2} - 5348x + 30048$

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	25	75	159	363	444	1023	789	2064	1305	2001	3171	522	1512	−3168	−3873	−25482	−39261	−2730	−79641	−57504	−72318	−104274	−70527	103116

$\operatorname{tr} a_n$ 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)[7]   # 110.6.a.h

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

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