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

Newform orbit 106.6.a.d

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q + 28q^{2} + 19q^{3} + 112q^{4} + 89q^{5} + 76q^{6} + 84q^{7} + 448q^{8} + 984q^{9} + 356q^{10} + 639q^{11} + 304q^{12} + 843q^{13} + 336q^{14} + 4764q^{15} + 1792q^{16} + 2426q^{17} + 3936q^{18} + 1457q^{19} + 1424q^{20} + 1608q^{21} + 2556q^{22} + 4308q^{23} + 1216q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{7} - 108x^{6} - 15833x^{5} + 1674030x^{4} + 57050161x^{3} - 6782736930x^{2} + 34476336939x + 2430502212540$

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
$\operatorname{tr} a_p$	28	19	89	84	639	843	2426	1457	4308	4469	16897	−4499	7102	−16345	4010	−19663	853	−20764	−52032	−6971	−58930	−132842	−72305	35485	−27826

$\operatorname{tr} a_n$ 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)[3]   # 106.6.a.d

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

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