Atlas › Modular forms › Weight 4 › Level 161 › 161.4.a › 161.4.a.b

Newform orbit 161.4.a.b

Trace form

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

$8q - 3q^{3} + 24q^{4} - 24q^{5} - 41q^{6} - 56q^{7} - 69q^{8} + 95q^{9} - 30q^{10} - 98q^{11} - 131q^{12} - 145q^{13} - 232q^{15} - 76q^{16} - 96q^{17} - 69q^{18} - 226q^{19} - 22q^{20} + 21q^{21} - 98q^{22} + 184q^{23} - 216q^{24} + O(q^{25})$

Coefficient field

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

$x^{8} + 3x^{7} - 154x^{6} + 7x^{5} + 4258x^{4} - 5641x^{3} - 4596x^{2} + 343x + 147$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
23	$-1$

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
$\operatorname{tr} a_p$	0	−3	−24	−56	−98	−145	−96	−226	184	73	−987	−2	−907	636	421	138	−924	−890	−1392	259	−297	−1788	2114	−1040	−1452

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.119 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, 7, 23, 161]: [1, 3, 7, 50] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 34 = 1 x 34 (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(161, 4)[1]   # 161.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 161:4 1000