Atlas › Modular forms › Weight 4 › Level 143 › 143.4.a › 143.4.a.b

Newform orbit 143.4.a.b

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - 6q^{2} - 6q^{3} + 26q^{4} - 8q^{5} - 15q^{6} - 53q^{7} - 36q^{8} - 52q^{10} + 66q^{11} - 19q^{12} - 78q^{13} - 120q^{14} - 23q^{15} + 26q^{16} - 117q^{17} - 27q^{18} - 67q^{19} + 10q^{20} + 19q^{21} - 66q^{22} - 158q^{23} - 609q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} + 12x^{5} - 31x^{4} - 363x^{3} - 750x^{2} - 511x - 106$

Atkin–Lehner signs

$p$	$w_p$
11	$-1$
13	$+1$

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
$\operatorname{tr} a_p$	−6	−6	−8	−53	66	−78	−117	−67	−158	−145	−58	−753	−232	−390	−205	−65	1735	421	−703	445	−2340	−1234	−1601	−442	−2682

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.116 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, 11, 13, 143]: [1, 4, 5, 44] (certified at ell = 2147482921)
T = [(2, 1), (3, 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(143, 4)[1]   # 143.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 143:4 1000