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

Newform orbit 195.4.a.g

Trace form

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

$3q + q^{2} + 9q^{3} + 25q^{4} + 15q^{5} + 3q^{6} - 12q^{7} + 87q^{8} + 27q^{9} + 5q^{10} + 10q^{11} + 75q^{12} - 39q^{13} - 188q^{14} + 45q^{15} + 305q^{16} - 54q^{17} + 9q^{18} + 34q^{19} + 125q^{20} - 36q^{21} - 202q^{22} + 252q^{23} + 261q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 11x^{2} - 315x - 81$

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
$\operatorname{tr} a_p$	1	9	15	−12	10	−39	−54	34	252	−76	128	388	432	500	236	1220	492	−992	−992	−202	1068	−3006	236	−1384	882

$\operatorname{tr} a_n$ 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)[6]   # 195.4.a.g

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

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