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

Newform orbit 161.4.a.d

Trace form

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

$12q + 4q^{2} + q^{3} + 72q^{4} + 16q^{5} - 15q^{6} + 84q^{7} - 21q^{8} + 203q^{9} + 106q^{10} + 50q^{11} - 139q^{12} + 21q^{13} + 28q^{14} + 192q^{15} + 516q^{16} + 26q^{17} + 251q^{18} + 292q^{19} - 12q^{20} + 7q^{21} + 206q^{22} + 276q^{23} + 42q^{24} + O(q^{25})$

Coefficient field

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

$x^{12} - 5x^{11} - 320x^{10} + 1493x^{9} + 32188x^{8} - 118513x^{7} - 1432230x^{6} + 3344891x^{5} + 29772725x^{4} - 30228170x^{3} - 252581562x^{2} + 83941312x + 730973278$

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
$\operatorname{tr} a_p$	4	1	16	84	50	21	26	292	276	129	375	958	−347	392	559	42	−620	−1258	2024	1371	−1241	2660	−1476	1670	1174

$\operatorname{tr} a_n$ 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)[3]   # 161.4.a.d

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

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