Atlas › Modular forms › Weight 2 › Level 815 › 815.2.a › 815.2.a.e

Newform orbit 815.2.a.e

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$16q + 7q^{2} - 6q^{3} + 21q^{4} + 16q^{5} + 4q^{6} + 3q^{7} + 12q^{8} + 26q^{9} + 7q^{10} + 19q^{11} - q^{13} + 6q^{14} - 6q^{15} + 19q^{16} - q^{17} - 6q^{18} + 9q^{19} + 21q^{20} + 4q^{21} - 29q^{22} + 14q^{23} + 2q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} - x^{15} - 67x^{14} + 72x^{13} + 1665x^{12} - 1693x^{11} - 20106x^{10} + 17880x^{9} + 126883x^{8} - 91398x^{7} - 413878x^{6} + 211350x^{5} + 639873x^{4} - 157591x^{3} - 394535x^{2} - 6300x + 35095$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
163	$+1$

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
$\operatorname{tr} a_p$	7	−6	16	3	19	−1	−1	9	14	47	−6	19	14	33	−50	36	1	24	29	−10	10	40	−34	20	−3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.339 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, 163, 815]: [0, 1, 14, 84] (certified at ell = 2147482693)
T = [(2, 1), (3, 1)]; h of degree 55 = 1 x 55 (phi(ord eps) x dim S_k^new); 6 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(815, 2)[4]   # 815.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 815:2 1000