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

Newform orbit 206.4.a.c

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 8q^{2} - 12q^{3} + 16q^{4} - 13q^{5} - 24q^{6} - 25q^{7} + 32q^{8} - 2q^{9} - 26q^{10} - 117q^{11} - 48q^{12} - 80q^{13} - 50q^{14} - 45q^{15} + 64q^{16} - 42q^{17} - 4q^{18} - 162q^{19} - 52q^{20} - 122q^{21} - 234q^{22} - 227q^{23} - 96q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{4} + 25x^{3} + 161x^{2} - 164x - 2672$

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
$\operatorname{tr} a_p$	8	−12	−13	−25	−117	−80	−42	−162	−227	−37	−236	−26	169	−370	−209	−17	790	376	−344	323	907	−662	624	458	−239

$\operatorname{tr} a_n$ 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)[2]   # 206.4.a.c

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

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