Atlas › Modular forms › Weight 6 › Level 23 › 23.6.a › 23.6.a.a

Newform orbit 23.6.a.a

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 4q^{2} - 20q^{3} + 16q^{4} - 58q^{5} - 230q^{6} - 282q^{7} - 360q^{8} + 121q^{9} - 156q^{10} + 136q^{11} + 620q^{12} - 1116q^{13} + 2016q^{14} + 750q^{15} + 128q^{16} - 896q^{17} + 4282q^{18} + 1654q^{19} + 1504q^{20} - 1670q^{21} + 1352q^{22} - 1587q^{23} + 3600q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 24x^{2} + 37x + 2$

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
$\operatorname{tr} a_p$	−4	−20	−58	−282	136	−1116	−896	1654	−1587	−844	−3020	8938	−12792	−16730	22500	17108	54176	−71324	−62960	98400	−81772	58224	9892	27542	−273672

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.214 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, 11] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 2 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, 6)[0]   # 23.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 23:6 1000