Atlas › Modular forms › Weight 2 › Level 479 › 479.2.a › 479.2.a.a

Newform orbit 479.2.a.a

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} + 5x^{7} - 3x^{6} - 34x^{5} - 11x^{4} + 47x^{3} + 43x^{2} + 12x + 1$

Atkin–Lehner signs

$p$	$w_p$
479	$+1$

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
$\operatorname{tr} a_p$	−2	−3	−4	−4	−5	−9	−4	−11	0	−9	−19	−21	−16	−9	3	7	6	−14	−3	1	−20	−12	10	−1	−21

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.195 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, 479]: [0, 41] (certified at ell = 2147482877)
T = [(2, 1), (3, 1)]; h of degree 40 = 1 x 40 (phi(ord eps) x dim S_k^new); 4 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(479, 2)[0]   # 479.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 479:2 1000