Atlas › Modular forms › Weight 2 › Level 399 › 399.2.a › 399.2.a.f

Newform orbit 399.2.a.f

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{5} - 23x^{4} - 7244x^{3} + 80226x^{2} + 12451365x + 79229023$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
7	$-1$
19	$-1$

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
$\operatorname{tr} a_p$	1	5	−2	5	−2	8	−2	5	2	0	8	2	2	20	−26	4	−4	10	10	10	10	14	−34	10	−16

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.241 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, 3, 7, 19, 21, 57, 133, 399]: [0, 1, 1, 2, 4, 8, 14, 56] (certified at ell = 2147483647)
T = [(2, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 19 = 1 x 19 (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(399, 2)[5]   # 399.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 399:2 1000