Atlas › Modular forms › Weight 2 › Level 493 › 493.2.a › 493.2.a.g

Newform orbit 493.2.a.g

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ , whose minimal polynomial is

$x^{8} - 47x^{7} - 653x^{6} + 53470x^{5} - 508099x^{4} - 4910096x^{3} + 62496389x^{2} + 130902805x - 1483158668$

Atkin–Lehner signs

$p$	$w_p$
17	$+1$
29	$-1$

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	6	1	3	10	3	−8	−13	20	8	−7	−6	3	9	−2	9	45	−19	−10	10	21	−10	49	−29	−13

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.309 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, 17, 29, 493]: [0, 2, 3, 46] (certified at ell = 2147480161)
T = [(2, 1), (3, 3), (5, 5), (7, 7)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 8 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(493, 2)[6]   # 493.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 493:2 1000