Atlas › Modular forms › Weight 2 › Level 195 › 195.2.a › 195.2.a.e

Newform orbit 195.2.a.e

Trace form

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

$3q - 3q^{3} + 8q^{4} - 3q^{5} + q^{7} + 6q^{8} + 3q^{9} + q^{11} - 8q^{12} + 3q^{13} - 6q^{14} + 3q^{15} + 26q^{16} - q^{17} + 6q^{19} - 8q^{20} - q^{21} - 6q^{22} - 7q^{23} - 6q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{7} + 5T_{11} + 7T_{17} + 9T_{19}$ , whose minimal polynomial is

$x^{3} - 55x^{2} - 5921x + 212263$

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 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
$\operatorname{tr} a_p$	0	−3	−3	1	1	3	−1	6	−7	18	6	13	1	0	−18	11	−8	9	4	−11	−6	5	8	11	−25

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.106 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 3, 5, 13, 15, 39, 65, 195]: [0, 1, 1, 1, 4, 6, 8, 32] (certified at ell = 2147483629)
T = [(2, 1), (7, 3), (11, 5), (17, 7), (19, 9)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 2 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, 2)[4]   # 195.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 195:2 1000