Atlas › Modular forms › Weight 4 › Level 55 › 55.4.a › 55.4.a.b

Newform orbit 55.4.a.b

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 7q^{2} - 3q^{3} + 17q^{4} + 10q^{5} + 19q^{6} - 25q^{7} - 63q^{8} - 41q^{9} - 35q^{10} - 22q^{11} - 85q^{12} - 50q^{13} + 11q^{14} - 15q^{15} + 297q^{16} - 151q^{17} + 118q^{18} - 3q^{19} + 85q^{20} - 39q^{21} + 77q^{22} + 48q^{23} + 307q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{2} + 16x - 4$

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	−3	10	−25	−22	−50	−151	−3	48	−221	141	−559	−144	−60	−48	117	−86	−155	266	1587	70	−1294	−558	−1777	−334

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.102 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, 4, 20] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 2 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, 4)[1]   # 55.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 55:4 1000