Atlas › Modular forms › Weight 2 › Level 143 › 143.2.a › 143.2.a.b

Newform orbit 143.2.a.b

Trace form

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

$4q + 3q^{2} + 3q^{4} - q^{6} + 6q^{7} + 9q^{8} + 2q^{9} - 8q^{10} + 4q^{11} + 4q^{12} - 4q^{13} - 4q^{14} - 10q^{15} + 5q^{16} + 6q^{17} - 15q^{18} + 8q^{19} - 24q^{20} - 2q^{21} + 3q^{22} - 4q^{23} + 2q^{24} + O(q^{25})$

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

$x^{4} - 3x^{3} - 7x^{2} + 29x - 21$

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
$\operatorname{tr} a_p$	3	0	0	6	4	−4	6	8	−4	−10	2	12	8	26	−18	−6	−16	−12	2	−14	22	−10	−2	10	22

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.172 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 11, 13, 143]: [0, 2, 1, 16] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 2 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, 2)[1]   # 143.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 143:2 1000