Atlas › Modular forms › Weight 2 › Level 167 › 167.2.a › 167.2.a.b

Newform orbit 167.2.a.b

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q + 2q^{2} + 3q^{3} + 14q^{4} + 4q^{5} - 6q^{6} + 11q^{7} + 3q^{8} + 17q^{9} - 3q^{10} - q^{12} + 9q^{13} - 7q^{14} - 3q^{15} + 10q^{16} + 3q^{17} - 5q^{18} - 11q^{20} - 16q^{21} + 8q^{22} + q^{23} - 31q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{12} - 11x^{11} - 179x^{10} + 2277x^{9} + 7507x^{8} - 143202x^{7} + 63800x^{6} + 2992752x^{5} - 5606187x^{4} - 13117270x^{3} + 24571265x^{2} + 17224374x - 7449939$

Atkin–Lehner signs

$p$	$w_p$
167	$-1$

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	4	11	0	9	3	0	1	−6	2	34	−14	6	2	10	−28	2	28	−9	61	0	−16	−12	73

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.164 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, 167]: [0, 15] (certified at ell = 2147483237)
T = [(2, 1), (3, 3)]; h of degree 14 = 1 x 14 (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(167, 2)[1]   # 167.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 167:2 1000