Atlas › Modular forms › Weight 2 › Level 573 › 573.2.a › 573.2.a.g

Newform orbit 573.2.a.g

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{10} - 4x^{9} - 48x^{8} + 177x^{7} + 801x^{6} - 2851x^{5} - 5344x^{4} + 19063x^{3} + 11040x^{2} - 40937x - 2026$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
191	$-1$

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	−10	1	11	8	8	−8	8	21	−7	7	18	−39	46	0	−6	−8	7	39	6	−12	12	0	−37	16

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.216 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, 191, 573]: [0, 1, 17, 66] (certified at ell = 2147483171)
T = [(2, 1), (5, 1)]; h of degree 31 = 1 x 31 (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(573, 2)[6]   # 573.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 573:2 1000