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

Newform orbit 55.6.a.b

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 5q^{2} + 61q^{4} - 100q^{5} - 157q^{6} - 90q^{7} - 135q^{8} + 22q^{9} + 125q^{10} - 484q^{11} - 795q^{12} + 820q^{13} - 1687q^{14} - 2671q^{16} - 3800q^{17} - 1610q^{18} - 3394q^{19} - 1525q^{20} - 4708q^{21} + 605q^{22} - 3020q^{23} + 2721q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} + 5x^{3} - 422x^{2} - 180x + 26696$

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
$\operatorname{tr} a_p$	−5	0	−100	−90	−484	820	−3800	−3394	−3020	−5248	4732	10210	−21068	−12140	4720	−21670	−69068	−44000	−8720	−47516	−2480	−188192	68620	−170266	186160

$\operatorname{tr} a_n$ 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)[1]   # 55.6.a.b

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

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