Atlas › Modular forms › Weight 2 › Level 597 › 597.2.a › 597.2.a.e

Newform orbit 597.2.a.e

Trace form

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

$14q + 2q^{2} + 14q^{3} + 20q^{4} + q^{5} + 2q^{6} + 11q^{7} + 3q^{8} + 14q^{9} + 4q^{10} + 20q^{12} + 15q^{13} - 4q^{14} + q^{15} + 28q^{16} + 5q^{17} + 2q^{18} + 19q^{19} - 11q^{20} + 11q^{21} + 6q^{23} + 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{14} - 5x^{13} - 451x^{12} + 2986x^{11} + 70521x^{10} - 560724x^{9} - 4592932x^{8} + 43302864x^{7} + 124417808x^{6} - 1544551121x^{5} - 713062996x^{4} + 24248009298x^{3} - 18955958278x^{2} - 120099054178x + 151519161847$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
199	$+1$

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	14	1	11	0	15	5	19	6	−14	11	1	−17	38	−5	−12	9	14	10	−19	37	11	45	−25	3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.221 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, 199, 597]: [0, 1, 17, 68] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; h of degree 33 = 1 x 33 (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(597, 2)[4]   # 597.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 597:2 1000