Atlas › Modular forms › Weight 2 › Level 491 › 491.2.a › 491.2.a.b

Newform orbit 491.2.a.b

Trace form

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

$10q - 3q^{2} - 4q^{3} + 3q^{4} - 9q^{5} - q^{6} - 6q^{7} - 3q^{8} + 2q^{9} - 13q^{10} - 3q^{11} - 8q^{12} - 15q^{13} + 4q^{14} + q^{15} - 7q^{16} - 3q^{17} + 4q^{18} - 23q^{19} - 28q^{21} - 8q^{22} - 3q^{23} - 10q^{24} + O(q^{25})$

Coefficient field

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

$x^{10} + 15x^{9} - 40x^{8} - 1294x^{7} - 1592x^{6} + 32607x^{5} + 71934x^{4} - 254211x^{3} - 590513x^{2} + 504838x + 1084676$

Atkin–Lehner signs

$p$	$w_p$
491	$+1$

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
$\operatorname{tr} a_p$	−3	−4	−9	−6	−3	−15	−3	−23	−3	−5	−12	−18	−18	0	27	2	−4	−62	−6	4	−12	−29	−19	28	−49

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.201 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, 491]: [0, 42] (certified at ell = 2147474491)
T = [(2, 1), (3, 3)]; h of degree 41 = 1 x 41 (phi(ord eps) x dim S_k^new); 6 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(491, 2)[1]   # 491.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 491:2 1000