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

Newform orbit 501.2.a.d

Trace form

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

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

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

$x^{8} + 2x^{7} - 41x^{6} - 31x^{5} + 531x^{4} - 15x^{3} - 2310x^{2} + 1210x + 1219$

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	1	0	5	0	−7	24	1	−11	30	11	10	24	−3	−25	45	16	18	21	−8	10	7	26	−3

$\operatorname{tr} a_n$ 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)[3]   # 501.2.a.d

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

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