Atlas › Modular forms › Weight 2 › Level 385 › 385.2.a › 385.2.a.e

Newform orbit 385.2.a.e

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 3q^{2} - 4q^{3} + 5q^{4} - 3q^{5} + 2q^{6} + 3q^{7} - 9q^{8} + 3q^{9} + 3q^{10} + 3q^{11} + 2q^{12} - 8q^{13} - 3q^{14} + 4q^{15} + 5q^{16} - 14q^{17} - 5q^{18} + 2q^{19} - 5q^{20} - 4q^{21} - 3q^{22} - 6q^{23} - 8q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 15x^{2} + 47x + 37$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
7	$-1$
11	$-1$

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
$\operatorname{tr} a_p$	−3	−4	−3	3	3	−8	−14	2	−6	2	−4	8	−10	−2	−26	4	−8	−8	−6	−20	−6	−8	10	−16	0

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.31 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, 7, 11, 35, 55, 77, 385]: [0, 1, 1, 2, 6, 8, 10, 52] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 19 = 1 x 19 (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(385, 2)[4]   # 385.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 385:2 1000