Atlas › Modular forms › Weight 6 › Level 106 › 106.6.a › 106.6.a.a

Newform orbit 106.6.a.a

Trace form

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

$4q + 16q^{2} - 8q^{3} + 64q^{4} - 136q^{5} - 32q^{6} - 210q^{7} + 256q^{8} + 12q^{9} - 544q^{10} - 450q^{11} - 128q^{12} - 1692q^{13} - 840q^{14} - 2678q^{15} + 1024q^{16} - 3200q^{17} + 48q^{18} - 2686q^{19} - 2176q^{20} - 3676q^{21} - 1800q^{22} + 1570q^{23} - 512q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{4} + 144x^{3} + 4062x^{2} - 81090x + 280835$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
53	$-1$

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
$\operatorname{tr} a_p$	16	−8	−136	−210	−450	−1692	−3200	−2686	1570	1816	−14498	−8548	−9896	3622	−3552	11236	−51300	−2768	98070	−72488	−10100	20176	49358	31564	−94776

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.281 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, 2, 53, 106]: [1, 2, 23, 69] (certified at ell = 2147483053)
T = [(3, 1), (5, 1)]; h of degree 23 = 1 x 23 (phi(ord eps) x dim S_k^new); 6 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(106, 6)[0]   # 106.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 106:6 1000