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

Newform orbit 543.2.a.c

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{7} + 13x^{6} - 48x^{5} - 800x^{4} + 652x^{3} + 10226x^{2} - 1797x - 23311$

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 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
$\operatorname{tr} a_p$	−4	−7	−3	−1	−4	−11	−23	6	−15	−8	1	−13	−30	8	−16	−11	−5	3	−5	7	−37	14	2	−14	−42

$\operatorname{tr} a_n$ 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)[2]   # 543.2.a.c

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

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