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

Newform orbit 106.4.a.a

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 4q^{2} - 8q^{3} + 8q^{4} - 16q^{5} - 16q^{6} - 28q^{7} + 16q^{8} - 16q^{9} - 32q^{10} - 44q^{11} - 32q^{12} - 74q^{13} - 56q^{14} + 52q^{15} + 32q^{16} - 30q^{17} - 32q^{18} + 16q^{19} - 64q^{20} + 64q^{21} - 88q^{22} + 36q^{23} - 64q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{3})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 24x + 141$

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
$\operatorname{tr} a_p$	4	−8	−16	−28	−44	−74	−30	16	36	−6	128	354	−548	380	−568	−106	−112	−632	464	276	−144	−116	56	−2156	−1794

$\operatorname{tr} a_n$ 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)[0]   # 106.4.a.a

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

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