Atlas › Modular forms › Weight 4 › Level 155 › 155.4.a › 155.4.a.e

Newform orbit 155.4.a.e

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q + 9q^{2} + 6q^{3} + 51q^{4} + 50q^{5} + 50q^{7} + 87q^{8} + 100q^{9} + 45q^{10} - 12q^{11} - 52q^{12} + 150q^{13} + 44q^{14} + 30q^{15} + 195q^{16} + 208q^{17} + 351q^{18} + 230q^{19} + 255q^{20} + 40q^{21} + 44q^{22} + 152q^{23} - 176q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} - 15x^{9} - 138x^{8} + 2394x^{7} + 6896x^{6} - 138888x^{5} - 157858x^{4} + 3451578x^{3} + 1886739x^{2} - 30779253x - 13200704$

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
$\operatorname{tr} a_p$	9	6	50	50	−12	150	208	230	152	156	310	936	150	−466	334	−588	−1158	640	1630	−458	1036	124	−842	−2484	2756

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.121 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, 31, 155]: [1, 3, 9, 50] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 30 = 1 x 30 (phi(ord eps) x dim S_k^new); 5 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, 4)[4]   # 155.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 155:4 1000