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

Newform orbit 165.4.a.g

Trace form

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

$3q + q^{2} - 9q^{3} + 17q^{4} + 15q^{5} - 3q^{6} + 6q^{7} - 3q^{8} + 27q^{9} + 5q^{10} + 33q^{11} - 51q^{12} - 20q^{13} + 144q^{14} - 45q^{15} + 25q^{16} + 32q^{17} + 9q^{18} + 116q^{19} + 85q^{20} - 18q^{21} + 11q^{22} + 240q^{23} + 9q^{24} + O(q^{25})$

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

$x^{3} - 19x^{2} - 2846x - 20876$

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 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
$\operatorname{tr} a_p$	1	−9	15	6	33	−20	32	116	240	238	92	−90	−46	−134	−220	−798	1236	342	764	1816	100	−96	858	838	−1322

$\operatorname{tr} a_n$ 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)[6]   # 165.4.a.g

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

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