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

Newform orbit 299.2.a.g

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q + q^{2} + 3q^{3} + 19q^{4} + 3q^{5} + q^{6} - 2q^{7} + 17q^{9} + 6q^{10} + 3q^{11} + 10q^{12} - 10q^{13} - 15q^{14} + 2q^{15} + 25q^{16} - 3q^{17} + q^{18} + 2q^{19} - 19q^{20} + 21q^{21} + 13q^{22} + 10q^{23} - 35q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{10} - 10x^{9} - 184x^{8} + 1872x^{7} + 10918x^{6} - 121792x^{5} - 173628x^{4} + 3089160x^{3} - 2523311x^{2} - 20669838x + 33020188$

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	3	3	−2	3	−10	−3	2	10	17	5	16	−16	−9	−11	8	2	48	−6	24	−33	17	−21	−16	−40

$\operatorname{tr} a_n$ 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)[6]   # 299.2.a.g

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

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