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

Newform orbit 155.4.a.d

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q - q^{2} + 4q^{3} + 39q^{4} - 45q^{5} + 20q^{6} + 20q^{7} - 33q^{8} + 175q^{9} + 5q^{10} + 4q^{11} + 282q^{12} + 188q^{13} - 56q^{14} - 20q^{15} + 467q^{16} - 86q^{17} - 189q^{18} + 150q^{19} - 195q^{20} + 340q^{21} + 316q^{22} - 182q^{23} + 530q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} - 3x^{8} - 280x^{7} + 592x^{6} + 25246x^{5} - 22774x^{4} - 847524x^{3} - 243820x^{2} + 9335981x + 12048357$

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	4	−45	20	4	188	−86	150	−182	442	−279	114	52	148	−156	470	302	2362	−540	−290	2870	2768	−84	1086	−1450

$\operatorname{tr} a_n$ 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)[3]   # 155.4.a.d

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

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