Atlas › Modular forms › Weight 2 › Level 398 › 398.2.a › 398.2.a.d

Newform orbit 398.2.a.d

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{6} - 5x^{5} - 40x^{4} + 215x^{3} + 230x^{2} - 2257x + 2651$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
199	$-1$

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
$\operatorname{tr} a_p$	−6	1	4	11	−7	−2	2	−2	1	−2	9	33	10	16	17	0	−31	−18	11	−10	10	−1	7	3	−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 0.289 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, 199, 398]: [0, 1, 17, 52] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 2 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(398, 2)[3]   # 398.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 398:2 1000