Atlas › Modular forms › Weight 2 › Level 807 › 807.2.a › 807.2.a.e

Newform orbit 807.2.a.e

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q + 4q^{2} + 13q^{3} + 14q^{4} + 5q^{5} + 4q^{6} - q^{7} + 9q^{8} + 13q^{9} - 3q^{10} + 15q^{11} + 14q^{12} + 11q^{13} + 2q^{14} + 5q^{15} - 4q^{16} + 36q^{17} + 4q^{18} + q^{19} + 18q^{20} - q^{21} + 4q^{22} + 6q^{23} + 9q^{24} + O(q^{25})$

Coefficient field

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

$x^{13} - 9x^{12} - 17x^{11} + 328x^{10} - 344x^{9} - 3285x^{8} + 6547x^{7} + 6587x^{6} - 17507x^{5} - 4807x^{4} + 15592x^{3} + 1452x^{2} - 3825x + 381$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
269	$+1$

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
$\operatorname{tr} a_p$	4	13	5	−1	15	11	36	1	6	18	−10	−23	29	−8	−13	15	13	−18	−8	16	12	−17	19	38	31

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.345 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, 269, 807]: [0, 1, 23, 92] (certified at ell = 2147482661)
T = [(2, 1), (5, 1)]; h of degree 45 = 1 x 45 (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(807, 2)[4]   # 807.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 807:2 1000