Atlas › Modular forms › Weight 2 › Level 591 › 591.2.a › 591.2.a.e

Newform orbit 591.2.a.e

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q - 14q^{3} + 20q^{4} + 7q^{7} + 14q^{9} + 8q^{10} + 4q^{11} - 20q^{12} + 8q^{13} + 6q^{14} + 40q^{16} + 4q^{17} + 15q^{19} - 6q^{20} - 7q^{21} + 12q^{22} - 3q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{14} - 87x^{12} + 17x^{11} + 2890x^{10} - 1113x^{9} - 45391x^{8} + 25855x^{7} + 334840x^{6} - 250313x^{5} - 949754x^{4} + 865765x^{3} + 249868x^{2} - 282756x + 9566$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
197	$-1$

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
$\operatorname{tr} a_p$	0	−14	0	7	4	8	4	15	−3	7	18	21	23	27	−19	12	−16	7	22	−4	18	2	−51	28	54

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.211 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, 197, 591]: [0, 1, 17, 68] (certified at ell = 2147479489)
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(591, 2)[4]   # 591.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 591:2 1000