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

Newform orbit 105.6.a.f

Trace form

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

$2q + 3q^{2} - 18q^{3} - 27q^{4} + 50q^{5} - 27q^{6} + 98q^{7} - 39q^{8} + 162q^{9} + 75q^{10} - 232q^{11} + 243q^{12} - 204q^{13} + 147q^{14} - 450q^{15} - 527q^{16} - 932q^{17} + 243q^{18} - 2552q^{19} - 675q^{20} - 882q^{21} + 172q^{22} - 6920q^{23} + 351q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{65})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{11}$ , whose minimal polynomial is

$x^{2} + 229x + 8414$

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
$\operatorname{tr} a_p$	3	−18	50	98	−232	−204	−932	−2552	−6920	916	−7080	−10988	964	6912	1240	−7748	36264	−39244	−62192	10840	19388	−39952	30376	154372	−58564

$\operatorname{tr} a_n$ 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)[5]   # 105.6.a.f

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

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