Atlas › Modular forms › Weight 4 › Level 167 › 167.4.a › 167.4.a.a

Newform orbit 167.4.a.a

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q - 7q^{2} - 13q^{3} + 41q^{4} - 26q^{5} - 20q^{6} - 85q^{7} - 84q^{8} + 28q^{9} - 63q^{10} - 39q^{11} - 207q^{12} - 142q^{13} - 69q^{14} - 114q^{15} - 83q^{16} - 154q^{17} - 224q^{18} - 215q^{19} - 161q^{20} - 168q^{21} - 564q^{22} - 150q^{23} - 205q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{15} + 20x^{14} - 77x^{13} - 3078x^{12} - 2482x^{11} + 156371x^{10} + 215309x^{9} - 3561590x^{8} - 3719741x^{7} + 37832097x^{6} + 22701406x^{5} - 185261047x^{4} - 25792078x^{3} + 354998638x^{2} - 108407162x - 74375019$

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 odd 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
$\operatorname{tr} a_p$	−7	−13	−26	−85	−39	−142	−154	−215	−150	−461	−449	−1454	−288	−584	151	278	250	79	−1906	936	−2500	−764	2436	−1411	−6311

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.124 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 engine's checks

dim M^+ at levels [1, 167]: [1, 43] (certified at ell = 2147483237)
T = [(2, 1), (3, 1)]; h of degree 41 = 1 x 41 (phi(ord eps) x dim S_k^new); 6 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, 4)[0]   # 167.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 167:4 1000