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

Newform orbit 493.2.a.h

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ , whose minimal polynomial is

$x^{10} + 80x^{9} + 1271x^{8} - 41991x^{7} - 1405399x^{6} - 8666494x^{5} + 102306002x^{4} + 1439029351x^{3} + 4542783606x^{2} - 6214239818x - 37555501649$

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 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	−9	−4	−4	−19	−1	−10	4	−22	−10	−14	−9	−12	−35	−20	−15	13	−8	12	9	−22	−5	−45	7	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 0.309 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)[7]   # 493.2.a.h

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

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