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

Newform orbit 143.4.a.a

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 4q^{3} + 6q^{4} - 6q^{5} - 13q^{6} - 17q^{7} + 6q^{8} - 44q^{9} - 82q^{10} - 44q^{11} + 17q^{12} + 52q^{13} - 40q^{14} + 9q^{15} - 142q^{16} - 41q^{17} - 75q^{18} - 41q^{19} - 174q^{20} - 179q^{21} - 136q^{23} - 171q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} + 4x^{3} - 30x^{2} + 15x + 4$

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
$\operatorname{tr} a_p$	0	−4	−6	−17	−44	52	−41	−41	−136	−207	−348	−333	−198	−252	561	343	−541	−801	−863	781	−1306	−278	437	2552	−2504

$\operatorname{tr} a_n$ 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)[0]   # 143.4.a.a

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

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