Atlas › Modular forms › Weight 4 › Level 206 › 206.4.a › 206.4.a.a

Newform orbit 206.4.a.a

Trace form

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

$2q - 4q^{2} + 8q^{4} - 14q^{5} - q^{7} - 16q^{8} + 50q^{9} + 28q^{10} + 2q^{11} - 37q^{13} + 2q^{14} - 52q^{15} + 32q^{16} - 25q^{17} - 100q^{18} - 129q^{19} - 56q^{20} - 78q^{21} - 4q^{22} + 277q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{13})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 14x + 36$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
103	$-1$

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	0	−14	−1	2	−37	−25	−129	277	−11	−348	−192	−27	−276	−316	124	267	221	−420	−548	1264	−865	1731	−1540	−1113

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.14 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, 103, 206]: [1, 2, 27, 80] (certified at ell = 2147482501)
T = [(3, 1), (5, 1)]; h of degree 26 = 1 x 26 (phi(ord eps) x dim S_k^new); 5 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(206, 4)[0]   # 206.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 206:4 1000