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

Newform orbit 205.4.a.f

Trace form

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

$11q + 6q^{2} + 8q^{3} + 52q^{4} - 55q^{5} + 5q^{6} - 52q^{7} + 27q^{8} + 143q^{9} - 30q^{10} + 128q^{11} + 347q^{12} + 26q^{13} + 317q^{14} - 40q^{15} + 644q^{16} + 226q^{17} + 83q^{18} - 6q^{19} - 260q^{20} + 66q^{21} - 67q^{22} + 182q^{23} + 168q^{24} + O(q^{25})$

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

$x^{11} - 14x^{10} - 197x^{9} + 2643x^{8} + 14873x^{7} - 160333x^{6} - 506631x^{5} + 3377597x^{4} + 4923938x^{3} - 22248105x^{2} - 4077808x + 312372$

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
$\operatorname{tr} a_p$	6	8	−55	−52	128	26	226	−6	182	74	570	−460	−451	−156	914	836	2580	58	−838	−194	288	−394	2582	4046	−1386

$\operatorname{tr} a_n$ 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)[5]   # 205.4.a.f

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

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