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

Newform orbit 655.2.a.d

Trace form

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

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

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

$x^{13} - 9x^{12} - 8x^{11} + 255x^{10} - 319x^{9} - 2312x^{8} + 4008x^{7} + 9464x^{6} - 14908x^{5} - 20725x^{4} + 14373x^{3} + 24042x^{2} + 9117x + 1085$

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
$\operatorname{tr} a_p$	4	5	13	2	0	12	21	3	14	0	−3	35	31	0	17	21	16	−14	12	16	13	−44	1	29	25

$\operatorname{tr} a_n$ 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)[3]   # 655.2.a.d

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

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