Atlas › Modular forms › Weight 2 › Level 797 › 797.2.a › 797.2.a.c

Newform orbit 797.2.a.c

Trace form

The 25 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$25q - 6q^{2} - 16q^{3} + 14q^{4} - 4q^{5} - 25q^{7} - 18q^{8} + 19q^{9} - 12q^{10} - 16q^{11} - 15q^{12} - 16q^{13} + 6q^{14} - 15q^{15} - 5q^{17} - 29q^{18} - 44q^{19} + q^{20} + 2q^{21} - 22q^{22} - 9q^{23} - 4q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 25. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{25} + 22x^{24} + 163x^{23} + 189x^{22} - 3510x^{21} - 15407x^{20} + 11607x^{19} + 195057x^{18} + 200865x^{17} - 1107633x^{16} - 2165101x^{15} + 3302944x^{14} + 9490776x^{13} - 5453893x^{12} - 22722862x^{11} + 5674827x^{10} + 31330491x^{9} - 5870520x^{8} - 23625899x^{7} + 6239971x^{6} + 7706154x^{5} - 3190757x^{4} - 127689x^{3} + 142418x^{2} - 3345x - 840$

Atkin–Lehner signs

$p$	$w_p$
797	$+1$

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
$\operatorname{tr} a_p$	−6	−16	−4	−25	−16	−16	−5	−44	−9	−9	−18	−13	10	−112	−6	−14	−6	−20	−85	−4	−18	−47	−14	26	−3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.369 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, 797]: [0, 67] (certified at ell = 2147483029)
T = [(2, 1), (3, 1)]; h of degree 66 = 1 x 66 (phi(ord eps) x dim S_k^new); 7 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(797, 2)[2]   # 797.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 797:2 1000