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

Newform orbit 209.4.a.d

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q + 4q^{2} + 3q^{3} + 74q^{4} + 10q^{5} + 61q^{6} + 73q^{7} + 12q^{8} + 160q^{9} + 165q^{11} - 63q^{12} + 43q^{13} + 107q^{14} - 16q^{15} + 478q^{16} + 73q^{17} - q^{18} + 285q^{19} + 30q^{20} + 207q^{21} + 44q^{22} + 355q^{23} + 769q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{15} - 7x^{14} - 416x^{13} + 3000x^{12} + 59420x^{11} - 407384x^{10} - 3832582x^{9} + 22709284x^{8} + 120679850x^{7} - 519056428x^{6} - 1902876560x^{5} + 4574274300x^{4} + 11882387433x^{3} - 15794071257x^{2} - 22356999886x + 11217989776$

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 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
$\operatorname{tr} a_p$	4	3	10	73	165	43	73	285	355	385	478	−2	366	732	332	−555	−653	884	665	380	1263	3604	−1420	−3856	−2472

$\operatorname{tr} a_n$ 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)[3]   # 209.4.a.d

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

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