Atlas › Modular forms › Weight 8 › Level 23 › 23.8.a › 23.8.a.a

Newform orbit 23.8.a.a

Trace form

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

$5q - 16q^{2} - 68q^{3} + 256q^{4} - 56q^{5} + 538q^{6} - 1156q^{7} - 5952q^{8} - 2195q^{9} - 11260q^{10} - 1318q^{11} - 28012q^{12} - 19662q^{13} - 6848q^{14} - 24600q^{15} + 32448q^{16} - 5002q^{17} - 24650q^{18} - 38314q^{19} + 104440q^{20} + 33280q^{21} - 74872q^{22} + 60835q^{23} + 361992q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} + 84x^{4} - 1828x^{3} - 185902x^{2} - 3013037x - 13403350$

Atkin–Lehner signs

$p$	$w_p$
23	$-1$

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
$\operatorname{tr} a_p$	−16	−68	−56	−1156	−1318	−19662	−5002	−38314	60835	−150634	−179940	−752672	−1192910	−932646	−1008460	897104	1020972	−2758364	−1523138	3044884	−8872022	−4437540	−4637362	6381402	−6432034

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.256 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, 23]: [1, 15] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 3 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(23, 8)[0]   # 23.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 23:8 1000