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

Newform orbit 23.8.a.b

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + 40q^{3} + 640q^{4} + 444q^{5} - 1745q^{6} + 1446q^{7} + 3177q^{8} + 13878q^{9} + 19502q^{10} + 7588q^{11} + 22733q^{12} + 19862q^{13} + 17544q^{14} - 12770q^{15} + 64336q^{16} + 42070q^{17} - 59129q^{18} + 1050q^{19} + 3364q^{20} - 7698q^{21} - 128220q^{22} - 97336q^{23} - 621188q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} - 40x^{7} - 13974x^{6} + 575357x^{5} + 60679903x^{4} - 2143447954x^{3} - 107780594736x^{2} + 2281129020221x + 76387113833190$

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 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
$\operatorname{tr} a_p$	0	40	444	1446	7588	19862	42070	1050	−97336	−102578	304172	286472	1324414	2052578	675556	203654	−748892	61822	3235604	−4951664	11019370	4202464	518568	4203864	18621134

$\operatorname{tr} a_n$ 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)[1]   # 23.8.a.b

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

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