Atlas › Modular forms › Weight 2 › Level 362 › 362.2.a › 362.2.a.e

Newform orbit 362.2.a.e

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q - 5q^{2} + 4q^{3} + 5q^{4} - 4q^{6} + 5q^{7} - 5q^{8} + 5q^{9} + 4q^{11} + 4q^{12} + 10q^{13} - 5q^{14} + 8q^{15} + 5q^{16} - 5q^{18} + 4q^{19} + 2q^{21} - 4q^{22} - 4q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{5} - 4x^{4} - 188x^{3} + 803x^{2} + 4325x - 7949$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
181	$-1$

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	4	0	5	4	10	0	4	0	−10	12	22	−10	18	8	−11	−2	−4	23	−10	−10	−8	−6	−26	−6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.195 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, 2, 181, 362]: [0, 1, 15, 47] (certified at ell = 2147482801)
T = [(3, 1), (5, 3)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 3 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(362, 2)[4]   # 362.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 362:2 1000