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

Newform orbit 479.2.a.b

Trace form

The 32 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$32q + 3q^{2} + 5q^{3} + 43q^{4} + 6q^{5} + 7q^{6} + 8q^{7} + 6q^{8} + 55q^{9} + 2q^{10} + 11q^{11} + 3q^{12} + 13q^{13} + q^{14} + 11q^{15} + 61q^{16} + 5q^{18} + 19q^{19} + q^{20} + 10q^{21} + 6q^{22} + q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 32. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{32} - 8x^{31} - 104x^{30} + 957x^{29} + 4228x^{28} - 48577x^{27} - 81959x^{26} + 1392943x^{25} + 570303x^{24} - 25295422x^{23} + 7329678x^{22} + 308331082x^{21} - 214638469x^{20} - 2601650507x^{19} + 2406842138x^{18} + 15412369331x^{17} - 15806504776x^{16} - 64251441694x^{15} + 65109425841x^{14} + 187190452708x^{13} - 165826086624x^{12} - 375101029632x^{11} + 238890398341x^{10} + 499959702602x^{9} - 142211793688x^{8} - 403862086316x^{7} - 42021772396x^{6} + 142171527973x^{5} + 68617794259x^{4} + 9029625271x^{3} - 471701802x^{2} - 109995678x + 2550717$

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 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	6	8	11	13	0	19	0	21	37	49	28	1	−17	−3	−8	36	19	13	58	48	−44	49	81

$\operatorname{tr} a_n$ 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)[1]   # 479.2.a.b

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

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