Atlas › Modular forms › Weight 2 › Level 481 › 481.2.a › 481.2.a.e

Newform orbit 481.2.a.e

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$11q + 3q^{2} + 5q^{3} + 11q^{4} + 2q^{5} - 6q^{6} + 6q^{8} + 14q^{9} - 7q^{10} + 11q^{11} + 9q^{12} + 11q^{13} + 12q^{14} + 11q^{15} + 27q^{16} + 6q^{17} + 26q^{18} - 6q^{19} + q^{20} - 8q^{21} - 4q^{22} + 34q^{23} - 30q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{11} - 18x^{10} - 48x^{9} + 2061x^{8} - 2032x^{7} - 81065x^{6} + 138457x^{5} + 1192066x^{4} - 2037683x^{3} - 4796859x^{2} + 4307477x + 2145467$

Atkin–Lehner signs

$p$	$w_p$
13	$-1$
37	$+1$

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	5	2	0	11	11	6	−6	34	17	19	−11	−5	−5	1	0	12	−22	−1	45	−20	12	−17	11	−28

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.199 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, 13, 37, 481]: [0, 1, 3, 44] (certified at ell = 2147483629)
T = [(2, 1), (3, 3)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 5 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(481, 2)[4]   # 481.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 481:2 1000