Atlas › Modular forms › Weight 2 › Level 542 › 542.2.a › 542.2.a.h

Newform orbit 542.2.a.h

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{6} - 6x^{5} - 262x^{4} + 2156x^{3} + 12879x^{2} - 157280x + 348134$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
271	$-1$

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
$\operatorname{tr} a_p$	−6	0	2	10	2	0	12	2	10	−2	10	8	−12	38	−4	−2	−26	8	16	−36	10	−6	10	4	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 0.308 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, 2, 271, 542]: [0, 1, 23, 70] (certified at ell = 2147482801)
T = [(3, 1), (5, 3)]; h of degree 23 = 1 x 23 (phi(ord eps) x dim S_k^new); 4 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(542, 2)[7]   # 542.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 542:2 1000