Atlas › Modular forms › Weight 2 › Level 443 › 443.2.a › 443.2.a.e

Newform orbit 443.2.a.e

Trace form

The 22 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$22q + q^{2} + 8q^{3} + 27q^{4} + 3q^{5} - 4q^{6} + 12q^{7} + 3q^{8} + 24q^{9} + 9q^{10} - 8q^{11} + 9q^{12} + 49q^{13} + 7q^{15} + 29q^{16} + 15q^{17} + 5q^{18} + 16q^{19} + 5q^{20} + 15q^{21} + q^{22} - 10q^{23} - 28q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 22. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{22} - 9x^{21} - 36x^{20} + 515x^{19} + 57x^{18} - 11644x^{17} + 13973x^{16} + 133029x^{15} - 253020x^{14} - 813220x^{13} + 1971022x^{12} + 2606520x^{11} - 7798367x^{10} - 4143410x^{9} + 16128887x^{8} + 3328032x^{7} - 17767694x^{6} - 1460634x^{5} + 9842103x^{4} + 324352x^{3} - 2216577x^{2} + 78971x + 123703$

Atkin–Lehner signs

$p$	$w_p$
443	$-1$

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
$\operatorname{tr} a_p$	1	8	3	12	−8	49	15	16	−10	3	13	51	−4	34	−10	17	−5	73	13	−12	45	−4	−6	−4	27

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.287 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, 443]: [0, 38] (certified at ell = 2147481311)
T = [(2, 1), (3, 1)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 4 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(443, 2)[4]   # 443.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 443:2 1000