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

Newform orbit 795.2.a.k

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q - q^{2} + 5q^{3} + 13q^{4} - 5q^{5} - q^{6} + 9q^{7} + 6q^{8} + 5q^{9} + q^{10} - 7q^{11} + 13q^{12} + 12q^{13} + q^{14} - 5q^{15} + 9q^{16} - 10q^{17} - q^{18} + 9q^{19} - 13q^{20} + 9q^{21} - 12q^{22} + 6q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} - 26x^{4} + 49x^{3} + 3432x^{2} - 29552x + 67776$

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 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	5	−5	9	−7	12	−10	9	0	2	6	−1	3	11	−11	−5	0	17	11	0	−14	9	30	36	7

$\operatorname{tr} a_n$ 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)[10]   # 795.2.a.k

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

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