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

Newform orbit 209.4.a.b

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q - 6q^{2} - 9q^{3} + 14q^{4} - 10q^{5} + q^{6} - 53q^{7} - 42q^{8} + 65q^{9} + 4q^{10} + 110q^{11} - q^{12} - 127q^{13} - 33q^{14} - 276q^{15} - 82q^{16} - 165q^{17} - 343q^{18} - 190q^{19} - 226q^{20} - 75q^{21} - 66q^{22} - 121q^{23} - 31q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} + 15x^{9} - 103x^{8} - 1913x^{7} + 3172x^{6} + 78946x^{5} - 3953x^{4} - 1075395x^{3} - 1027633x^{2} + 1568551x + 1360688$

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
$\operatorname{tr} a_p$	−6	−9	−10	−53	110	−127	−165	−190	−121	−645	−920	−184	−982	−398	612	−39	595	−542	−1287	−550	549	−2592	598	−1062	2734

$\operatorname{tr} a_n$ 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)[1]   # 209.4.a.b

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

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