Atlas › Modular forms › Weight 2 › Level 802 › 802.2.a › 802.2.a.e

Newform orbit 802.2.a.e

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{9} + 11x^{8} + 22x^{7} - 79x^{6} - 177x^{5} + 175x^{4} + 358x^{3} - 122x^{2} - 192x - 28$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
401	$+1$

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
$\operatorname{tr} a_p$	−9	−6	−5	−7	−2	1	−16	6	−17	−6	−8	−8	−13	−15	−47	−31	−26	11	−15	−21	1	−13	−20	−12	−13

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.247 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, 401, 802]: [0, 1, 34, 102] (certified at ell = 2147482801)
T = [(3, 1), (5, 1)]; h of degree 33 = 1 x 33 (phi(ord eps) x dim S_k^new); 4 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(802, 2)[4]   # 802.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 802:2 1000