Atlas › Modular forms › Weight 4 › Level 183 › 183.4.a › 183.4.a.e

Newform orbit 183.4.a.e

Trace form

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

$11q + 5q^{2} + 33q^{3} + 59q^{4} + 15q^{5} + 15q^{6} + 52q^{7} + 75q^{8} + 99q^{9} + 101q^{10} + 28q^{11} + 177q^{12} + 148q^{13} + 8q^{14} + 45q^{15} + 311q^{16} + 219q^{17} + 45q^{18} + 56q^{19} - 227q^{20} + 156q^{21} + 276q^{22} - 131q^{23} + 225q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{11} - 20x^{10} - 997x^{9} + 24378x^{8} + 218825x^{7} - 7875942x^{6} - 3062649x^{5} + 954793624x^{4} - 2177864476x^{3} - 42930143624x^{2} + 95305316736x + 628479259296$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
61	$-1$

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
$\operatorname{tr} a_p$	5	33	15	52	28	148	219	56	−131	158	652	589	448	855	416	486	−786	671	202	−1277	135	1344	−679	199	2823

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.129 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, 3, 61, 183]: [1, 2, 17, 64] (certified at ell = 2147482921)
T = [(2, 1), (5, 1)]; h of degree 30 = 1 x 30 (phi(ord eps) x dim S_k^new); 5 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(183, 4)[4]   # 183.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 183:4 1000