Atlas › Modular forms › Weight 6 › Level 105 › 105.6.a › 105.6.a.e

Newform orbit 105.6.a.e

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + q^{2} - 18q^{3} + 53q^{4} - 50q^{5} - 9q^{6} - 98q^{7} + 111q^{8} + 162q^{9} - 25q^{10} - 196q^{11} - 477q^{12} + 32q^{13} - 49q^{14} + 450q^{15} - 2223q^{16} + 932q^{17} + 81q^{18} - 1164q^{19} - 1325q^{20} + 882q^{21} - 2428q^{22} + 1832q^{23} - 999q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{233})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{11}$ , whose minimal polynomial is

$x^{2} + 195x - 11522$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
5	$+1$
7	$+1$

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
$\operatorname{tr} a_p$	1	−18	−50	−98	−196	32	932	−1164	1832	−1108	−13740	−8244	2620	15472	−9016	24752	−38224	−14868	65464	34244	−59696	−150648	−82832	136284	11064

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.337 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 engine's checks

dim M^+ at levels [1, 3, 5, 7, 15, 21, 35, 105]: [1, 3, 3, 5, 12, 16, 22, 84] (certified at ell = 2147483629)
T = [(2, 1), (11, 1)]; h of degree 20 = 1 x 20 (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(105, 6)[4]   # 105.6.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 105:6 1000