Atlas › Modular forms › Weight 4 › Level 106 › 106.4.a › 106.4.a.b

Newform orbit 106.4.a.b

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 6q^{2} - 5q^{3} + 12q^{4} - 6q^{5} + 10q^{6} + 8q^{7} - 24q^{8} + 30q^{9} + 12q^{10} - 100q^{11} - 20q^{12} + 7q^{13} - 16q^{14} - 144q^{15} + 48q^{16} - 157q^{17} - 60q^{18} - 153q^{19} - 24q^{20} - 270q^{21} + 200q^{22} - 117q^{23} + 40q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 11x^{2} + 11x - 77$

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
$\operatorname{tr} a_p$	−6	−5	−6	8	−100	7	−157	−153	−117	−325	138	−29	−138	574	−6	159	−1270	954	−10	−1697	1178	221	−757	−1434	443

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.109 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, 15, 43] (certified at ell = 2147483053)
T = [(3, 1), (5, 1)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 3 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, 4)[1]   # 106.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 106:4 1000