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

Newform orbit 183.4.a.d

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + 3q^{2} - 24q^{3} + 23q^{4} + 25q^{5} - 9q^{6} + 10q^{7} + 21q^{8} + 72q^{9} - 27q^{10} + 44q^{11} - 69q^{12} + 92q^{13} + 36q^{14} - 75q^{15} + 191q^{16} + 239q^{17} + 27q^{18} + 36q^{19} + 603q^{20} - 30q^{21} + 108q^{22} + 321q^{23} - 63q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{8} - 28x^{7} - 416x^{6} + 13952x^{5} + 26677x^{4} - 1614054x^{3} + 1486850x^{2} + 2455628x + 457936$

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
$\operatorname{tr} a_p$	3	−24	25	10	44	92	239	36	321	478	−198	141	820	−621	52	−394	−170	−488	−1000	123	1667	−2718	3795	1699	2181

$\operatorname{tr} a_n$ 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)[3]   # 183.4.a.d

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

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