Atlas › Modular forms › Weight 2 › Level 501 › 501.2.a › 501.2.a.e

Newform orbit 501.2.a.e

Trace form

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

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

Coefficient field

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

$x^{8} - 10x^{7} + 21x^{6} + 69x^{5} - 305x^{4} + 303x^{3} + 122x^{2} - 336x + 133$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
167	$+1$

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	7	−4	13	0	11	12	7	1	−2	−9	4	2	17	9	29	−12	0	13	−20	8	33	4	−31

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.293 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, 167, 501]: [0, 1, 15, 58] (certified at ell = 2147483237)
T = [(2, 1), (5, 1)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 3 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(501, 2)[4]   # 501.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 501:2 1000