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

Newform orbit 155.4.a.b

Trace form

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

$4q + 2q^{2} - 4q^{3} + 6q^{4} - 20q^{5} - 18q^{6} - 24q^{7} + 30q^{8} + 46q^{9} - 10q^{10} - 82q^{11} - 118q^{12} - 50q^{13} - 128q^{14} + 20q^{15} - 166q^{16} + 26q^{17} - 130q^{18} - 86q^{19} - 30q^{20} - 212q^{21} - 62q^{22} + 24q^{23} - 54q^{24} + O(q^{25})$

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

$x^{4} + 2x^{3} - 76x^{2} + 74x + 507$

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 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
$\operatorname{tr} a_p$	2	−4	−20	−24	−82	−50	26	−86	24	−334	124	−640	−34	−436	1060	228	−174	−814	−512	−838	674	−112	−84	−692	−756

$\operatorname{tr} a_n$ 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)[1]   # 155.4.a.b

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

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