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

Newform orbit 161.4.a.a

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q - 4q^{2} - 11q^{3} - 4q^{5} + 35q^{7} + 18q^{8} + 14q^{9} + 26q^{10} - 36q^{11} - 28q^{12} - 69q^{13} - 28q^{14} - 88q^{15} - 124q^{16} - 42q^{17} - 94q^{18} - 140q^{19} - 168q^{20} - 77q^{21} - 346q^{22} - 115q^{23} + 18q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{5} + 15x^{4} + 18x^{3} - 312x^{2} + 73x + 453$

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
$\operatorname{tr} a_p$	−4	−11	−4	35	−36	−69	−42	−140	−115	−383	−319	−154	277	−718	53	450	936	−522	−810	−1373	459	−728	−500	−586	−2150

$\operatorname{tr} a_n$ 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)[0]   # 161.4.a.a

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

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