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

Newform orbit 105.6.a.g

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 8q^{2} - 36q^{3} + 114q^{4} - 100q^{5} - 72q^{6} + 196q^{7} + 216q^{8} + 324q^{9} - 200q^{10} + 232q^{11} - 1026q^{12} - 304q^{13} + 392q^{14} + 900q^{15} + 5266q^{16} + 1512q^{17} + 648q^{18} + 2632q^{19} - 2850q^{20} - 1764q^{21} + 11692q^{22} - 4128q^{23} - 1944q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{11}$ , whose minimal polynomial is

$x^{4} - 240x^{3} - 658613x^{2} + 109138580x + 83192094636$

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
$\operatorname{tr} a_p$	8	−36	−100	196	232	−304	1512	2632	−4128	6952	5352	−3272	2840	−2448	−25728	3328	−47296	28712	98656	37400	48464	140976	−106368	−74152	−166112

$\operatorname{tr} a_n$ 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)[6]   # 105.6.a.g

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

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