Atlas › Modular forms › Weight 2 › Level 466 › 466.2.a › 466.2.a.e

Newform orbit 466.2.a.e

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{5} + 5x^{4} - x^{3} - 38x^{2} - 60x - 25$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
233	$+1$

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	0	−5	−5	−12	5	−12	−9	−7	−9	−16	4	−33	7	−12	5	−18	5	18	−9	16	3	−3	−15	30

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.197 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, 2, 233, 466]: [0, 1, 20, 60] (certified at ell = 2147482481)
T = [(3, 1), (5, 1)]; h of degree 19 = 1 x 19 (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(466, 2)[4]   # 466.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 466:2 1000