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

Newform orbit 165.4.a.h

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 4q^{2} + 12q^{3} + 26q^{4} + 20q^{5} + 12q^{6} + 34q^{7} + 48q^{8} + 36q^{9} + 20q^{10} - 44q^{11} + 78q^{12} + 2q^{13} - 52q^{14} + 60q^{15} + 66q^{16} + 74q^{17} + 36q^{18} + 136q^{19} + 130q^{20} + 102q^{21} - 44q^{22} - 64q^{23} + 144q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{7}$ , whose minimal polynomial is

$x^{4} - 106x^{3} + 1803x^{2} + 86294x - 2064924$

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
$\operatorname{tr} a_p$	4	12	20	34	−44	2	74	136	−64	52	492	−4	268	546	−276	−184	−1032	116	−552	−920	926	1152	−134	−1064	−1648

$\operatorname{tr} a_n$ 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)[7]   # 165.4.a.h

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

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