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

Newform orbit 209.4.a.c

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q - 2q^{2} + 11q^{3} + 82q^{4} + 8q^{5} + 13q^{6} + 39q^{7} + 6q^{8} + 156q^{9} + 124q^{10} - 143q^{11} + 247q^{12} - 23q^{13} + 47q^{14} + 278q^{15} + 526q^{16} + 73q^{17} - 165q^{18} - 247q^{19} + 142q^{20} + 77q^{21} + 22q^{22} + 185q^{23} - 251q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{13} - 9x^{12} - 319x^{11} + 2439x^{10} + 36869x^{9} - 216377x^{8} - 2090825x^{7} + 8555183x^{6} + 61792009x^{5} - 159968289x^{4} - 892800811x^{3} + 1339569583x^{2} + 4930561818x - 3821607640$

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
$\operatorname{tr} a_p$	−2	11	8	39	−143	−23	73	−247	185	−41	930	1094	−766	1024	516	1017	15	−862	621	632	−629	2444	732	3178	1410

$\operatorname{tr} a_n$ 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)[2]   # 209.4.a.c

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

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