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

Newform orbit 143.4.a.d

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$11q + 6q^{2} + 6q^{3} + 66q^{4} - 4q^{5} - 14q^{6} + 45q^{7} + 78q^{8} + 135q^{9} + 48q^{10} + 121q^{11} + 105q^{12} + 143q^{13} - 48q^{14} - 125q^{15} + 394q^{16} + 265q^{17} + 405q^{18} + 127q^{19} - 46q^{20} - 287q^{21} + 66q^{22} + 42q^{23} - 83q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{11} - 12x^{10} - 207x^{9} + 2151x^{8} + 17358x^{7} - 131460x^{6} - 699398x^{5} + 3247241x^{4} + 11800157x^{3} - 35275384x^{2} - 67069355x + 137663052$

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
$\operatorname{tr} a_p$	6	6	−4	45	121	143	265	127	42	435	−174	187	128	696	−355	−693	−609	1625	633	−1937	404	1670	785	1464	1184

$\operatorname{tr} a_n$ 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)[3]   # 143.4.a.d

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

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