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

Newform orbit 481.2.a.b

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{7} + 2x^{6} - 94x^{5} - 231x^{4} + 2447x^{3} + 5741x^{2} - 19679x - 38651$

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 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	1	−4	−8	−13	−7	2	−8	−18	−9	−25	−7	7	−3	−3	−4	−26	8	5	−61	−14	−12	5	−31	2

$\operatorname{tr} a_n$ 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)[1]   # 481.2.a.b

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

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