Atlas › Modular forms › Weight 2 › Level 227 › 227.2.a › 227.2.a.e

Newform orbit 227.2.a.e

Trace form

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

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

Coefficient field

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

$x^{10} - x^{9} - 37x^{8} + 24x^{7} + 473x^{6} - 218x^{5} - 2390x^{4} + 1349x^{3} + 4078x^{2} - 3704x + 706$

Atkin–Lehner signs

$p$	$w_p$
227	$-1$

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
$\operatorname{tr} a_p$	0	1	7	0	−3	23	17	−16	−16	−3	14	38	8	12	−14	7	−12	13	18	−14	33	13	−5	18	39

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.17 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, 227]: [0, 20] (certified at ell = 2147479573)
T = [(2, 1), (3, 1)]; h of degree 19 = 1 x 19 (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(227, 2)[4]   # 227.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 227:2 1000