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

Newform orbit 55.6.a.c

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 9q^{2} + 115q^{4} + 125q^{5} - 237q^{6} + 70q^{7} + 753q^{8} + 1059q^{9} + 225q^{10} - 605q^{11} - 1605q^{12} + 1498q^{13} + 2113q^{14} + 4883q^{16} + 3874q^{17} + 5838q^{18} + 882q^{19} + 2875q^{20} + 1092q^{21} - 1089q^{22} - 5344q^{23} - 13119q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{5} - 9x^{4} - 997x^{3} + 3073x^{2} + 255296x + 991236$

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
$\operatorname{tr} a_p$	9	0	125	70	−605	1498	3874	882	−5344	5318	−7916	−1788	5854	−4364	46452	4412	17896	−35930	73136	43612	142306	−46504	81604	8664	−22230

$\operatorname{tr} a_n$ 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)[2]   # 55.6.a.c

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

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