Atlas › Modular forms › Weight 2 › Level 809 › 809.2.a › 809.2.a.b

Newform orbit 809.2.a.b

Trace form

The 24 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$24q - 4q^{2} - 11q^{3} + 14q^{4} - 10q^{5} - 12q^{6} - 19q^{7} - 6q^{8} + 15q^{9} - 10q^{10} - 12q^{11} - 22q^{12} - 14q^{13} - 15q^{14} - 14q^{15} - 2q^{16} - 9q^{17} + 5q^{18} - 45q^{19} - 4q^{20} - 11q^{21} - 15q^{22} - 13q^{23} - 13q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 24. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{24} + 15x^{23} + 50x^{22} - 282x^{21} - 1961x^{20} + 351x^{19} + 24497x^{18} + 27763x^{17} - 150480x^{16} - 277948x^{15} + 499685x^{14} + 1247126x^{13} - 873250x^{12} - 3062664x^{11} + 623293x^{10} + 4292247x^{9} + 195840x^{8} - 3412545x^{7} - 531796x^{6} + 1491755x^{5} + 251734x^{4} - 329971x^{3} - 30240x^{2} + 29057x - 2417$

Atkin–Lehner signs

$p$	$w_p$
809	$+1$

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
$\operatorname{tr} a_p$	−4	−11	−10	−19	−12	−14	−9	−45	−13	−15	−72	−19	−20	−19	−15	23	−3	−43	−9	−29	5	−106	38	1	−18

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.263 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, 809]: [0, 68] (certified at ell = 2147474929)
T = [(2, 1), (3, 1)]; h of degree 67 = 1 x 67 (phi(ord eps) x dim S_k^new); 7 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(809, 2)[1]   # 809.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 809:2 1000