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

Newform orbit 106.6.a.b

Trace form

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

$6q - 24q^{2} - 17q^{3} + 96q^{4} - 86q^{5} + 68q^{6} + 74q^{7} - 384q^{8} + 455q^{9} + 344q^{10} - 330q^{11} - 272q^{12} + 347q^{13} - 296q^{14} + 2130q^{15} + 1536q^{16} + 279q^{17} - 1820q^{18} - 1353q^{19} - 1376q^{20} - 5032q^{21} + 1320q^{22} - 1329q^{23} + 1088q^{24} + O(q^{25})$

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

$x^{6} + 103x^{5} - 8240x^{4} - 588246x^{3} + 2588603x^{2} + 336521963x + 2872066508$

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
$\operatorname{tr} a_p$	−24	−17	−86	74	−330	347	279	−1353	−1329	−21557	−15884	−11365	−30904	−8744	−36130	−16854	−17010	−5830	11136	7383	126090	14873	4247	−139644	127379

$\operatorname{tr} a_n$ 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)[1]   # 106.6.a.b

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

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