Atlas › Modular forms › Weight 2 › Level 395 › 395.2.a › 395.2.a.h

Newform orbit 395.2.a.h

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$11q - 2q^{3} + 20q^{4} - 11q^{5} + 3q^{6} + 7q^{7} - 3q^{8} + 21q^{9} + 6q^{11} - 9q^{12} + 14q^{13} + 10q^{14} + 2q^{15} + 38q^{16} - 16q^{17} + 14q^{18} + 11q^{19} - 20q^{20} + 19q^{21} + 3q^{22} - 6q^{23} - 13q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{11} + 2x^{10} - 49x^{9} - 93x^{8} + 778x^{7} + 1450x^{6} - 4092x^{5} - 8096x^{4} + 1985x^{3} + 5404x^{2} + 185x - 523$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
79	$-1$

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
$\operatorname{tr} a_p$	0	−2	−11	7	6	14	−16	11	−6	5	15	38	4	4	−4	1	−11	23	45	7	−11	11	−5	4	19

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.194 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, 79, 395]: [0, 1, 7, 42] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 27 = 1 x 27 (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(395, 2)[7]   # 395.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 395:2 1000