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

Newform orbit 55.4.a.c

Trace form

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

$3q + 5q^{2} - 3q^{3} + 23q^{4} - 15q^{5} + 59q^{6} - 15q^{7} + 33q^{8} + 72q^{9} - 25q^{10} - 33q^{11} + 5q^{12} - 14q^{13} + 111q^{14} + 15q^{15} - 149q^{16} + 91q^{17} - 256q^{18} - 19q^{19} - 115q^{20} - 39q^{21} - 55q^{22} + 330q^{23} - 213q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 4x^{2} - 881x + 2558$

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
$\operatorname{tr} a_p$	5	−3	−15	−15	−33	−14	91	−19	330	413	−127	−543	58	532	478	261	−630	1095	−976	−445	−1838	318	−330	2573	−2744

$\operatorname{tr} a_n$ 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)[2]   # 55.4.a.c

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

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