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

Newform orbit 183.4.a.b

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 2q^{2} + 9q^{3} - 2q^{4} - 28q^{5} - 6q^{6} - 4q^{7} + 18q^{8} + 27q^{9} - 22q^{10} + 30q^{11} - 6q^{12} - 92q^{13} - 62q^{14} - 84q^{15} - 142q^{16} - 104q^{17} - 18q^{18} - 228q^{19} + 50q^{20} - 12q^{21} - 6q^{22} + 74q^{23} + 54q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{3} + 30x^{2} + 274x + 692$

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 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
$\operatorname{tr} a_p$	−2	9	−28	−4	30	−92	−104	−228	74	36	−422	46	−232	−818	−48	−68	−458	−183	−580	880	−468	−824	1836	−312	−266

$\operatorname{tr} a_n$ 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)[1]   # 183.4.a.b

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

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