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

Newform orbit 206.4.a.e

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 18q^{2} + 6q^{3} + 36q^{4} + 17q^{5} + 12q^{6} + 45q^{7} + 72q^{8} + 133q^{9} + 34q^{10} + 59q^{11} + 24q^{12} + 102q^{13} + 90q^{14} + 255q^{15} + 144q^{16} + 26q^{17} + 266q^{18} + 218q^{19} + 68q^{20} + 298q^{21} + 118q^{22} + 371q^{23} + 48q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{9} - 23x^{8} - 947x^{7} + 27404x^{6} + 92777x^{5} - 7551817x^{4} + 59423649x^{3} - 35519796x^{2} - 695069064x + 516713160$

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 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
$\operatorname{tr} a_p$	18	6	17	45	59	102	26	218	371	9	180	−104	−83	458	−13	−45	−142	−206	474	−667	−2291	−336	−1296	−1476	445

$\operatorname{tr} a_n$ 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)[4]   # 206.4.a.e

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

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