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

Newform orbit 55.6.a.a

Trace form

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

$3q - 7q^{2} - 36q^{3} + 41q^{4} + 75q^{5} + 101q^{6} - 102q^{7} - 15q^{8} + 249q^{9} - 175q^{10} + 363q^{11} - 1237q^{12} - 1646q^{13} - 963q^{14} - 900q^{15} - 2687q^{16} - 1742q^{17} - 3076q^{18} - 10q^{19} + 1025q^{20} + 1236q^{21} - 847q^{22} - 3876q^{23} + 5055q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 43x^{2} + 266x - 272$

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 odd 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
$\operatorname{tr} a_p$	−7	−36	75	−102	363	−1646	−1742	−10	−3876	1970	6596	−22032	−7214	29644	−8432	−31656	−31440	−49374	−111892	−58444	5554	76080	44844	25460	−150482

$\operatorname{tr} a_n$ 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)[0]   # 55.6.a.a

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

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