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

Newform orbit 443.2.a.d

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q - 3q^{2} - 7q^{3} + 11q^{4} - 7q^{5} - 8q^{6} - 16q^{7} - 15q^{8} + 9q^{9} - 15q^{10} - 2q^{11} - 15q^{12} - 42q^{13} + 8q^{14} + q^{15} + q^{16} - 5q^{17} + q^{18} - 5q^{20} - 19q^{21} - 15q^{22} - 6q^{23} + 8q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{12} + 10x^{11} + 18x^{10} - 85x^{9} - 265x^{8} + 49x^{7} + 659x^{6} + 340x^{5} - 373x^{4} - 345x^{3} - 67x^{2} + 3x + 1$

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 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
$\operatorname{tr} a_p$	−3	−7	−7	−16	−2	−42	−5	0	−6	−5	1	−46	4	−16	15	−35	4	−35	−5	13	−53	2	20	15	−29

$\operatorname{tr} a_n$ 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)[3]   # 443.2.a.d

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

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