Atlas › Modular forms › Weight 2 › Level 499 › 499.2.a › 499.2.a.b

Newform orbit 499.2.a.b

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$16q - 5q^{2} - 5q^{3} + 15q^{4} - 20q^{5} - 18q^{6} - 4q^{7} - 15q^{8} + 5q^{9} + q^{10} - 18q^{11} - q^{12} - 14q^{13} - 17q^{14} + q^{15} + 13q^{16} - 25q^{17} - 6q^{18} - 20q^{19} - 50q^{20} - 38q^{21} + 10q^{22} - 15q^{23} - 24q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} + 10x^{15} + 18x^{14} - 96x^{13} - 292x^{12} + 312x^{11} + 1397x^{10} - 403x^{9} - 3020x^{8} + 280x^{7} + 3107x^{6} - 404x^{5} - 1326x^{4} + 354x^{3} + 77x^{2} - 19x - 1$

Atkin–Lehner signs

$p$	$w_p$
499	$+1$

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
$\operatorname{tr} a_p$	−5	−5	−20	−4	−18	−14	−25	−20	−15	−40	−16	−11	−51	4	23	−26	−19	−38	20	−36	17	−2	0	−58	4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.196 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, 499]: [0, 42] (certified at ell = 2147475601)
T = [(2, 1), (3, 1)]; h of degree 41 = 1 x 41 (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(499, 2)[1]   # 499.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 499:2 1000