Atlas › Modular forms › Weight 4 › Level 195 › 195.4.a › 195.4.a.e

Newform orbit 195.4.a.e

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + q^{2} + 6q^{3} - 7q^{4} - 10q^{5} + 3q^{6} - 20q^{7} - 3q^{8} + 18q^{9} - 5q^{10} - 14q^{11} - 21q^{12} - 26q^{13} - 78q^{14} - 30q^{15} - 39q^{16} - 82q^{17} + 9q^{18} - 140q^{19} + 35q^{20} - 60q^{21} - 92q^{22} - 54q^{23} - 9q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{7}$ , whose minimal polynomial is

$x^{2} + 19x - 118$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
5	$+1$
13	$+1$

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	6	−10	−20	−14	−26	−82	−140	−54	−114	42	−288	−264	364	−694	−12	66	−556	−214	−1526	906	1860	−342	−924	2282

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.161 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, 13, 15, 39, 65, 195]: [1, 2, 3, 5, 8, 16, 24, 88] (certified at ell = 2147483629)
T = [(2, 1), (7, 1)]; h of degree 24 = 1 x 24 (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(195, 4)[4]   # 195.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 195:4 1000