Atlas › Modular forms › Weight 2 › Level 655 › 655.2.a › 655.2.a.e

Newform orbit 655.2.a.e

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{13} - 11x^{12} + 12x^{11} + 211x^{10} - 445x^{9} - 1708x^{8} + 3676x^{7} + 7322x^{6} - 12852x^{5} - 16347x^{4} + 19477x^{3} + 15342x^{2} - 10335x - 2843$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
131	$-1$

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	6	−13	−1	12	−5	25	9	8	12	7	−33	56	−15	21	−2	27	3	−20	10	−1	16	17	98	9

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.219 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, 5, 131, 655]: [0, 1, 12, 68] (certified at ell = 2147482481)
T = [(2, 1), (3, 1)]; h of degree 43 = 1 x 43 (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(655, 2)[4]   # 655.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 655:2 1000