Atlas › Modular forms › Weight 4 › Level 205 › 205.4.a › 205.4.a.e

Newform orbit 205.4.a.e

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$11q + 4q^{2} + 22q^{3} + 48q^{4} + 55q^{5} + 25q^{6} + 74q^{7} + 51q^{8} + 143q^{9} + 20q^{10} + 76q^{11} - 21q^{12} + 108q^{13} - 49q^{14} + 110q^{15} - 20q^{16} + 184q^{17} + 265q^{18} + 154q^{19} + 240q^{20} - 46q^{21} + 39q^{22} + 278q^{23} + 36q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{11} - 26x^{10} + 25x^{9} + 4075x^{8} - 24063x^{7} - 179595x^{6} + 1646263x^{5} + 1210265x^{4} - 33274754x^{3} + 29958573x^{2} + 210852720x - 294644300$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
41	$-1$

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	22	55	74	76	108	184	154	278	306	30	224	451	548	1396	−66	988	54	1952	−190	−2588	−1914	778	−538	1640

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.139 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, 5, 41, 205]: [1, 3, 12, 66] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 40 = 1 x 40 (phi(ord eps) x dim S_k^new); 6 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(205, 4)[4]   # 205.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 205:4 1000