Atlas › Modular forms › Weight 2 › Level 665 › 665.2.a › 665.2.a.k

Newform orbit 665.2.a.k

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{8} + 7x^{7} - 156x^{6} - 846x^{5} + 7245x^{4} + 29663x^{3} - 118970x^{2} - 270868x + 674808$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
7	$-1$
19	$-1$

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
$\operatorname{tr} a_p$	2	−3	8	8	16	2	−7	8	7	6	6	3	−7	23	0	−11	4	18	4	13	−1	10	−20	−2	−42

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.257 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, 7, 19, 35, 95, 133, 665]: [0, 1, 1, 2, 6, 12, 14, 84] (certified at ell = 2147483629)
T = [(2, 1), (3, 3)]; h of degree 35 = 1 x 35 (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(665, 2)[10]   # 665.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 665:2 1000