Atlas › Modular forms › Weight 2 › Level 795 › 795.2.a › 795.2.a.i

Newform orbit 795.2.a.i

Trace form

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

$4q - 4q^{2} - 4q^{3} + 6q^{4} + 4q^{5} + 4q^{6} + q^{7} - 15q^{8} + 4q^{9} - 4q^{10} + 3q^{11} - 6q^{12} - 12q^{13} - 9q^{14} - 4q^{15} + 18q^{16} - 14q^{17} - 4q^{18} - 9q^{19} + 6q^{20} - q^{21} - 10q^{23} + 15q^{24} + O(q^{25})$

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

$x^{4} + x^{3} - 92x^{2} - 340x + 8$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
5	$-1$
53	$-1$

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
$\operatorname{tr} a_p$	−4	−4	4	1	3	−12	−14	−9	−10	−10	6	−11	−17	−11	−17	4	4	−23	3	4	−26	19	−16	−10	13

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.309 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, 15, 53, 159, 265, 795]: [0, 1, 1, 4, 5, 20, 28, 112] (certified at ell = 2147483053)
T = [(2, 1), (7, 3)]; h of degree 35 = 1 x 35 (phi(ord eps) x dim S_k^new); 6 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(795, 2)[8]   # 795.2.a.i

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

cargo run --release -p sagebrush-web --example atlas -- 795:2 1000