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

Newform orbit 105.6.a.c

Trace form

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

$2q - 4q^{2} - 18q^{3} + 8q^{4} + 50q^{5} + 36q^{6} - 98q^{7} - 144q^{8} + 162q^{9} - 100q^{10} - 176q^{11} - 72q^{12} + 692q^{13} + 196q^{14} - 450q^{15} - 1248q^{16} - 428q^{17} - 324q^{18} - 1824q^{19} + 200q^{20} + 882q^{21} + 6752q^{22} + 8032q^{23} + 1296q^{24} + O(q^{25})$

Coefficient field

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

$x^{2} + 180x - 318332$

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 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
$\operatorname{tr} a_p$	−4	−18	50	−98	−176	692	−428	−1824	8032	−2948	14760	156	−5980	25672	21904	−53948	26296	−16788	16264	22264	28684	−64368	75848	32964	242604

$\operatorname{tr} a_n$ 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)[2]   # 105.6.a.c

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

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