Atlas › Modular forms › Weight 2 › Level 599 › 599.2.a › 599.2.a.b

Newform orbit 599.2.a.b

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{11} + 4x^{10} - 10x^{9} - 46x^{8} + 28x^{7} + 147x^{6} - 53x^{5} - 130x^{4} + 86x^{3} - 5x^{2} - 6x + 1$

Atkin–Lehner signs

$p$	$w_p$
599	$+1$

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 odd 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	−1	−2	−8	−5	−20	−16	−21	−3	−21	−3	−2	−14	−23	−2	17	−17	−40	16	10	−44	−40	38	−25	−27

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.218 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, 599]: [0, 51] (certified at ell = 2147480791)
T = [(2, 1), (3, 1)]; h of degree 50 = 1 x 50 (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(599, 2)[1]   # 599.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 599:2 1000