Atlas › Modular forms › Weight 4 › Level 209 › 209.4.a › 209.4.a.a

Newform orbit 209.4.a.a

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q - q^{3} + 22q^{4} - 12q^{5} + q^{6} - 59q^{7} - 36q^{8} - 19q^{9} - 120q^{10} - 88q^{11} - 55q^{12} - 61q^{13} - 93q^{14} - 62q^{15} - 34q^{16} + 107q^{17} - 183q^{18} + 152q^{19} - 82q^{20} - 145q^{21} - 259q^{23} - 315q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} + x^{7} - 142x^{6} + 202x^{5} + 4030x^{4} - 19056x^{3} + 34001x^{2} - 26701x + 7508$

Atkin–Lehner signs

$p$	$w_p$
11	$+1$
19	$-1$

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
$\operatorname{tr} a_p$	0	−1	−12	−59	−88	−61	107	152	−259	61	−276	−204	−402	−598	−892	−147	−1193	−1216	589	−538	−367	−2368	−918	2212	−3136

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.138 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, 11, 19, 209]: [1, 4, 6, 62] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 46 = 1 x 46 (phi(ord eps) x dim S_k^new); 7 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(209, 4)[0]   # 209.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 209:4 1000