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

Newform orbit 385.2.a.h

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{4} - 8x^{3} - 78x^{2} + 596x + 241$

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 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
$\operatorname{tr} a_p$	2	2	4	−4	−4	−8	−6	6	14	−4	10	−2	−10	−14	16	2	26	−12	−10	32	−14	20	−10	−10	−2

$\operatorname{tr} a_n$ 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)[7]   # 385.2.a.h

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

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