Atlas › Modular forms › Weight 2 › Level 662 › 662.2.a › 662.2.a.e

Newform orbit 662.2.a.e

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11}$ , whose minimal polynomial is

$x^{10} - 95x^{9} + 1497x^{8} + 117175x^{7} - 4877020x^{6} + 40658774x^{5} + 847684182x^{4} - 16794797054x^{3} + 78506834288x^{2} - 15304176387x - 281393088443$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
331	$+1$

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
$\operatorname{tr} a_p$	10	6	0	8	7	8	4	21	−5	−11	7	−2	−1	4	3	−16	8	−15	12	−4	−8	−17	−2	−1	26

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.264 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, 2, 331, 662]: [0, 1, 28, 85] (certified at ell = 2147482921)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 7 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(662, 2)[4]   # 662.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 662:2 1000