Atlas › Modular forms › Weight 2 › Level 643 › 643.2.a › 643.2.a.c

Newform orbit 643.2.a.c

Trace form

The 28 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$28q + 14q^{2} + 5q^{3} + 32q^{4} + 23q^{5} + 2q^{6} + 11q^{7} + 36q^{8} + 29q^{9} + 4q^{10} + 19q^{11} - 2q^{12} + 25q^{13} - 6q^{15} + 36q^{16} + 24q^{17} + 16q^{18} + q^{19} + 18q^{20} - 3q^{21} - 7q^{22} + 24q^{23} - q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 28. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{28} - 19x^{27} + 78x^{26} + 698x^{25} - 6628x^{24} + 3103x^{23} + 142573x^{22} - 399134x^{21} - 1166624x^{20} + 6246761x^{19} + 1952546x^{18} - 46961816x^{17} + 30921337x^{16} + 204445209x^{15} - 246496517x^{14} - 553958930x^{13} + 881577628x^{12} + 972418365x^{11} - 1822848179x^{10} - 1149512388x^{9} + 2286338794x^{8} + 961699176x^{7} - 1697940005x^{6} - 583090578x^{5} + 671548142x^{4} + 230063233x^{3} - 101710074x^{2} - 40612680x - 1982664$

Atkin–Lehner signs

$p$	$w_p$
643	$-1$

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
$\operatorname{tr} a_p$	14	5	23	11	19	25	24	1	24	54	−1	32	25	6	22	71	10	12	5	17	1	−5	3	22	−2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.218 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, 643]: [0, 54] (certified at ell = 2147476519)
T = [(2, 1), (3, 1)]; h of degree 53 = 1 x 53 (phi(ord eps) x dim S_k^new); 6 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(643, 2)[2]   # 643.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 643:2 1000