Atlas › Modular forms › Weight 10 › Level 23 › 23.10.a › 23.10.a.b

Newform orbit 23.10.a.b

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q + 32q^{2} + 235q^{3} + 3072q^{4} + 112q^{5} + 10975q^{6} + 1280q^{7} - 6519q^{8} + 51681q^{9} + 62294q^{10} - 2690q^{11} + 404069q^{12} + 307847q^{13} + 643288q^{14} + 585190q^{15} + 1684240q^{16} + 827614q^{17} + 2746471q^{18} + 475454q^{19} + 90204q^{20} + 2218032q^{21} + 2355076q^{22} + 2798410q^{23} + 6596308q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} - 267x^{9} - 103682x^{8} + 26353971x^{7} + 3516518700x^{6} - 803320657895x^{5} - 44713801674326x^{4} + 8700567080461209x^{3} + 171415439287597043x^{2} - 30934047659899510490x + 104637151790735845176$

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
$\operatorname{tr} a_p$	32	235	112	1280	−2690	307847	827614	475454	2798410	−6012071	−8628841	−4928252	26206047	39105260	90595459	174250464	94151060	174839396	−148457006	245537391	380918201	388717158	1453704440	−509637702	4762815042

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.301 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, 19] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 5 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, 10)[1]   # 23.10.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 23:10 1000