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

Newform orbit 143.4.a.c

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 8q^{3} + 46q^{4} + 30q^{5} + 34q^{6} + 25q^{7} + 36q^{8} + 91q^{9} - 22q^{10} - 99q^{11} + 181q^{12} - 117q^{13} + 351q^{15} + 130q^{16} + 53q^{17} + 33q^{18} + 69q^{19} + 282q^{20} + 463q^{21} + 216q^{23} - 121q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} - 8x^{8} - 228x^{7} + 1469x^{6} + 18148x^{5} - 89189x^{4} - 548824x^{3} + 2275194x^{2} + 5356779x - 20357814$

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 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
$\operatorname{tr} a_p$	0	8	30	25	−99	−117	53	69	216	−91	636	967	−226	42	−269	1227	−613	427	−271	2279	3602	−1182	−1877	1258	4002

$\operatorname{tr} a_n$ 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)[2]   # 143.4.a.c

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

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