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

Newform orbit 106.6.a.c

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - 24q^{2} + 10q^{3} + 96q^{4} + 39q^{5} - 40q^{6} - 24q^{7} - 384q^{8} + 698q^{9} - 156q^{10} + 759q^{11} + 160q^{12} - 498q^{13} + 96q^{14} - 1712q^{15} + 1536q^{16} - 723q^{17} - 2792q^{18} + 1002q^{19} + 624q^{20} - 1496q^{21} - 3036q^{22} + 3339q^{23} - 640q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{6} - 49x^{5} - 10932x^{4} + 527832x^{3} + 18282693x^{2} - 960677775x + 9530914230$

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 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
$\operatorname{tr} a_p$	−24	10	39	−24	759	−498	−723	1002	3339	20496	19809	11700	34212	6153	23484	16854	77427	−98496	−59334	62856	−101502	−61767	123084	82839	35085

$\operatorname{tr} a_n$ 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)[2]   # 106.6.a.c

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

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