Atlas › Modular forms › Weight 2 › Level 541 › 541.2.a › 541.2.a.b

Newform orbit 541.2.a.b

Trace form

The 24 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$24q + 3q^{2} + 12q^{3} + 23q^{4} + 13q^{5} + 3q^{6} + 2q^{7} + 3q^{8} + 26q^{9} + 38q^{11} + 11q^{12} - 3q^{13} + 16q^{14} + 16q^{15} + 17q^{16} + 14q^{17} - 5q^{18} + 24q^{19} + 27q^{20} + 2q^{21} - 8q^{22} + 15q^{23} - 2q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 24. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{24} - 15x^{23} + 25x^{22} + 670x^{21} - 3356x^{20} - 7598x^{19} + 81291x^{18} - 40040x^{17} - 848951x^{16} + 1363815x^{15} + 4602215x^{14} - 11030071x^{13} - 14041686x^{12} + 45651880x^{11} + 25650428x^{10} - 110928997x^{9} - 31893401x^{8} + 163803892x^{7} + 34820968x^{6} - 142985306x^{5} - 34004996x^{4} + 66259308x^{3} + 20057265x^{2} - 12208382x - 4548403$

Atkin–Lehner signs

$p$	$w_p$
541	$-1$

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	12	13	2	38	−3	14	24	15	17	17	−9	31	2	19	8	100	−12	−5	65	−22	−16	34	52	−11

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.204 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, 541]: [0, 45] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 44 = 1 x 44 (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(541, 2)[1]   # 541.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 541:2 1000