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

Newform orbit 55.4.a.a

$q$ -expansion

$q + q^{2} - 3q^{3} - 7q^{4} - 5q^{5} - 3q^{6} - 9q^{7} - 15q^{8} - 18q^{9} - 5q^{10} + 11q^{11} + 21q^{12} + 2q^{13} - 9q^{14} + 15q^{15} + 41q^{16} + 21q^{17} - 18q^{18} - 85q^{19} + 35q^{20} + 27q^{21} + 11q^{22} + 22q^{23} + 45q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{3}$ is a root of

$x + 8$

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
$\operatorname{tr} a_p$	1	−3	−5	−9	11	2	21	−85	22	−165	−83	1	−478	−8	126	−683	−290	257	776	−313	902	830	842	25	−1784

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 55, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

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)[0]   # 55.4.a.a

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

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