Atlas › Modular forms › Weight 2 › Level 155 › 155.2.a › 155.2.a.e

Newform orbit 155.2.a.e

Trace form

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

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

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

$x^{4} - 2x^{3} - 6x^{2} + 12x - 1$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
31	$+1$

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
$\operatorname{tr} a_p$	1	1	4	2	−4	10	11	−3	−2	−8	−4	3	−11	17	−10	13	−3	22	−12	−21	19	−2	−15	−10	4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.178 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 5, 31, 155]: [0, 1, 3, 18] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 11 = 1 x 11 (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(155, 2)[4]   # 155.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 155:2 1000