Atlas › Modular forms › Weight 2 › Level 229 › 229.2.a › 229.2.a.c

Newform orbit 229.2.a.c

Trace form

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

$11q + 5q^{2} + 3q^{3} + 11q^{4} + q^{7} + 15q^{8} + 14q^{9} - 3q^{10} + 27q^{11} - q^{12} - 7q^{13} + 4q^{14} - q^{15} + 7q^{16} + q^{17} - 4q^{18} + 8q^{19} - 16q^{20} - 7q^{21} + 4q^{22} - 2q^{23} - 16q^{24} + O(q^{25})$

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

$x^{11} - 8x^{10} - 8x^{9} + 201x^{8} - 297x^{7} - 1275x^{6} + 3286x^{5} + 1001x^{4} - 7152x^{3} + 1967x^{2} + 3270x + 50$

Atkin–Lehner signs

$p$	$w_p$
229	$-1$

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
$\operatorname{tr} a_p$	5	3	0	1	27	−7	1	8	−2	17	−3	−14	18	−2	14	−11	52	−23	18	31	−10	1	37	4	−17

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.252 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, 229]: [0, 19] (certified at ell = 2147478649)
T = [(2, 1), (3, 1)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 2 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(229, 2)[2]   # 229.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 229:2 1000