Atlas › Modular forms › Weight 6 › Level 55 › 55.6.a › 55.6.a.d

Newform orbit 55.6.a.d

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 3q^{2} + 115q^{4} - 150q^{5} + 325q^{6} - 66q^{7} + 249q^{8} + 832q^{9} - 75q^{10} + 726q^{11} + 1301q^{12} - 972q^{13} + 2017q^{14} + 6675q^{16} + 712q^{17} + 5920q^{18} + 610q^{19} - 2875q^{20} + 2508q^{21} + 363q^{22} + 7100q^{23} + 13447q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} - 3x^{5} - 1619x^{4} - 2391x^{3} + 703544x^{2} + 2119012x - 71685216$

Atkin–Lehner signs

$p$	$w_p$
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
$\operatorname{tr} a_p$	3	0	−150	−66	726	−972	712	610	7100	10964	8868	−15506	160	7032	−46320	30518	120532	−9228	47432	163948	78528	−36712	133792	254946	17404

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.238 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, 5, 11, 55]: [1, 3, 6, 32] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 18 = 1 x 18 (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(55, 6)[3]   # 55.6.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 55:6 1000