Atlas › Modular forms › Weight 2 › Level 805 › 805.2.a › 805.2.a.k

Newform orbit 805.2.a.k

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{11} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{5} + 116x^{4} + 4308x^{3} + 44832x^{2} - 440629x - 7597009$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
7	$+1$
23	$-1$

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
$\operatorname{tr} a_p$	−3	−6	5	−5	−11	−7	1	2	5	−14	−6	−1	7	−12	−24	−21	−1	7	−35	−32	−7	18	−1	1	−9

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.402 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, 7, 23, 35, 115, 161, 805]: [0, 1, 1, 3, 6, 14, 18, 100] (certified at ell = 2147483053)
T = [(2, 1), (3, 3), (11, 5), (13, 7), (17, 9)]; h of degree 43 = 1 x 43 (phi(ord eps) x dim S_k^new); 11 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(805, 2)[10]   # 805.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 805:2 1000