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

Newform orbit 161.4.a.c

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 9q^{3} + 48q^{4} - 4q^{5} + 46q^{6} - 63q^{7} + 66q^{8} + 122q^{9} + 50q^{10} - 8q^{11} + 220q^{12} + 25q^{13} + 88q^{15} + 180q^{16} - 28q^{17} - 54q^{18} + 254q^{19} + 302q^{20} - 63q^{21} - 122q^{22} - 207q^{23} + 624q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} - 9x^{8} - 248x^{7} + 1842x^{6} + 21679x^{5} - 120041x^{4} - 799840x^{3} + 2960374x^{2} + 10668992x - 21456414$

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 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
$\operatorname{tr} a_p$	0	9	−4	−63	−8	25	−28	254	−207	25	1171	70	221	−42	159	−774	1080	686	−370	1035	1979	2336	130	−1328	−104

$\operatorname{tr} a_n$ 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)[2]   # 161.4.a.c

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

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