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

Newform orbit 167.4.a.b

Trace form

The 26 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$26q + 5q^{2} + 11q^{3} + 129q^{4} + 34q^{5} + 28q^{6} + 83q^{7} + 60q^{8} + 325q^{9} + 77q^{10} + 93q^{11} + 81q^{12} + 222q^{13} + 99q^{14} + 186q^{15} + 621q^{16} + 288q^{17} + 316q^{18} + 165q^{19} + 439q^{20} + 168q^{21} + 404q^{22} + 402q^{23} + 371q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 26. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{26} - 16x^{25} - 582x^{24} + 9630x^{23} + 143044x^{22} - 2434790x^{21} - 20052747x^{20} + 343764145x^{19} + 1821609445x^{18} - 30118931779x^{17} - 115021552550x^{16} + 1709155753627x^{15} + 5227183587210x^{14} - 63282460715201x^{13} - 170963052018428x^{12} + 1495393282134301x^{11} + 3881508106120314x^{10} - 21295611658780743x^{9} - 57100116623466120x^{8} + 162241488567297709x^{7} + 492320470725463832x^{6} - 480826735624486696x^{5} - 2141618186449861476x^{4} - 443666870058802386x^{3} + 3243673881937847784x^{2} + 3106979012025371639x + 782927812525957665$

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
$\operatorname{tr} a_p$	5	11	34	83	93	222	288	165	402	3	419	1654	778	822	507	−810	−416	1311	2456	−824	2730	388	1566	365	7637

$\operatorname{tr} a_n$ 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)[1]   # 167.4.a.b

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

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