Atlas › Modular forms › Weight 2 › Level 579 › 579.2.a › 579.2.a.g

Newform orbit 579.2.a.g

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{13} - 8x^{12} - 32x^{11} + 397x^{10} - 228x^{9} - 5443x^{8} + 11772x^{7} + 15487x^{6} - 64085x^{5} + 34519x^{4} + 38200x^{3} - 27748x^{2} - 820x + 1228$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
193	$+1$

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
$\operatorname{tr} a_p$	2	13	6	15	3	11	−2	9	−8	5	25	29	−11	8	−12	14	10	6	25	0	8	7	−28	7	26

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.309 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, 3, 193, 579]: [0, 1, 16, 66] (certified at ell = 2147483137)
T = [(2, 1), (5, 1)]; h of degree 33 = 1 x 33 (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(579, 2)[6]   # 579.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 579:2 1000