Atlas › Modular forms › Weight 2 › Level 299 › 299.2.a › 299.2.a.e

Newform orbit 299.2.a.e

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{2} - 4x - 13$

Atkin–Lehner signs

$p$	$w_p$
13	$-1$
23	$+1$

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
$\operatorname{tr} a_p$	1	1	1	2	5	2	−12	5	−2	4	−4	10	−12	−2	−16	14	−16	10	−7	8	6	−18	−7	10	−13

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.184 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, 13, 23, 299]: [0, 1, 3, 30] (certified at ell = 2147483053)
T = [(2, 1), (3, 3)]; h of degree 23 = 1 x 23 (phi(ord eps) x dim S_k^new); 3 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(299, 2)[4]   # 299.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 299:2 1000