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

Newform orbit 23.10.a.a

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q - 89q^{3} + 1536q^{4} - 2388q^{5} - 12518q^{6} - 9896q^{7} + 34920q^{8} + 33064q^{9} - 60820q^{10} - 78484q^{11} - 343492q^{12} - 296769q^{13} - 711120q^{14} - 237870q^{15} - 253440q^{16} - 1128820q^{17} - 499874q^{18} - 1301252q^{19} - 3482704q^{20} - 108908q^{21} - 1562088q^{22} - 1958887q^{23} - 4606464q^{24} + O(q^{25})$

Coefficient field

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

$x^{7} + 89x^{6} - 71504x^{5} - 5546334x^{4} + 1158288801x^{3} + 55414954997x^{2} - 4635970367970x - 144096569656160$

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
$\operatorname{tr} a_p$	0	−89	−2388	−9896	−78484	−296769	−1128820	−1301252	−1958887	2813849	7334751	−13324320	−15691573	−46474818	8232227	−53545400	−341275144	−277157656	89654580	−286098961	−637495039	274469546	1164579762	−504153000	−3519929016

$\operatorname{tr} a_n$ 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)[0]   # 23.10.a.a

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

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