Atlas › Modular forms › Weight 4 › Level 165 › 165.4.a › 165.4.a.f

Newform orbit 165.4.a.f

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + q^{2} - 9q^{3} - 5q^{4} + 15q^{5} - 3q^{6} - 16q^{7} - 3q^{8} + 27q^{9} + 5q^{10} - 33q^{11} + 15q^{12} - 42q^{13} - 76q^{14} - 45q^{15} - 85q^{16} - 34q^{17} + 9q^{18} - 280q^{19} - 25q^{20} + 48q^{21} - 11q^{22} - 112q^{23} + 9q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{7}$ , whose minimal polynomial is

$x^{3} + 47x^{2} - 6597x - 172951$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
5	$-1$
11	$+1$

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
$\operatorname{tr} a_p$	1	−9	15	−16	−33	−42	−34	−280	−112	−290	−392	570	−662	−68	264	−94	−612	−582	940	−1616	738	124	−1232	838	−90

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.15 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, 3, 5, 11, 15, 33, 55, 165]: [1, 2, 3, 4, 8, 14, 20, 76] (certified at ell = 2147482921)
T = [(2, 1), (7, 3)]; h of degree 20 = 1 x 20 (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(165, 4)[5]   # 165.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 165:4 1000