Atlas › Modular forms › Weight 2 › Level 543 › 543.2.a › 543.2.a.e

Newform orbit 543.2.a.e

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{8} + 3x^{7} - 315x^{6} - 146x^{5} + 27972x^{4} - 47236x^{3} - 661641x^{2} + 2121659x - 1682491$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
181	$+1$

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	8	−2	−2	−2	8	12	8	0	12	−16	20	14	16	0	−22	20	2	16	−28	32	24	0	14	6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.307 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, 3, 181, 543]: [0, 1, 15, 62] (certified at ell = 2147482801)
T = [(2, 1), (5, 3)]; h of degree 31 = 1 x 31 (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(543, 2)[4]   # 543.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 543:2 1000