Atlas › Modular forms › Weight 2 › Level 593 › 593.2.a › 593.2.a.e

Newform orbit 593.2.a.e

Trace form

The 27 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$27q + 6q^{2} + 4q^{3} + 36q^{4} - q^{6} + 25q^{7} + 9q^{8} + 21q^{9} - 7q^{10} + 18q^{11} - 2q^{12} + 30q^{13} + 6q^{14} + 42q^{15} + 50q^{16} - 8q^{17} + 12q^{18} + 11q^{19} - 10q^{20} - 20q^{21} + 21q^{22} + 27q^{23} + 13q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 27. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{27} - 10x^{26} - 45x^{25} + 729x^{24} - 11x^{23} - 20850x^{22} + 32354x^{21} + 302710x^{20} - 722805x^{19} - 2454715x^{18} + 7363467x^{17} + 11782258x^{16} - 41160388x^{15} - 36892120x^{14} + 133247690x^{13} + 90609661x^{12} - 250557870x^{11} - 185843187x^{10} + 247370815x^{9} + 253712426x^{8} - 68843042x^{7} - 158094214x^{6} - 52384006x^{5} + 7626956x^{4} + 7915166x^{3} + 1703338x^{2} + 150920x + 4802$

Atkin–Lehner signs

$p$	$w_p$
593	$-1$

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
$\operatorname{tr} a_p$	6	4	0	25	18	30	−8	11	27	5	24	43	−31	30	2	26	−7	−19	92	9	12	63	−23	−15	5

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.222 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, 593]: [0, 50] (certified at ell = 2147470529)
T = [(2, 1), (3, 1)]; h of degree 49 = 1 x 49 (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(593, 2)[4]   # 593.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 593:2 1000