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

Newform orbit 543.2.a.d

Trace form

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

$8q + 3q^{2} - 8q^{3} + 5q^{4} + 5q^{5} - 3q^{6} - 3q^{7} + 6q^{8} + 8q^{9} + 2q^{10} + 4q^{11} - 5q^{12} + 13q^{13} + 5q^{14} - 5q^{15} + 3q^{16} + 27q^{17} + 3q^{18} - 10q^{19} + 21q^{20} + 3q^{21} + 11q^{22} + 3q^{23} - 6q^{24} + O(q^{25})$

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

$x^{8} - 18x^{7} - 93x^{6} + 2970x^{5} - 8872x^{4} - 54734x^{3} + 199387x^{2} + 273646x - 1006255$

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	5	−3	4	13	27	−10	3	16	−1	3	20	−4	12	23	7	1	−25	11	23	−26	16	14	28

$\operatorname{tr} a_n$ 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)[3]   # 543.2.a.d

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

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