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

Newform orbit 23.6.a.b

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 4q^{2} + 16q^{3} + 112q^{4} + 42q^{5} + 367q^{6} + 300q^{7} + 393q^{8} + 762q^{9} - 10q^{10} - 58q^{11} - 907q^{12} + 792q^{13} - 2984q^{14} - 1160q^{15} - 1904q^{16} - 400q^{17} - 2981q^{18} + 2738q^{19} - 7124q^{20} - 4848q^{21} - 1972q^{22} + 3174q^{23} - 3644q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} - 20x^{5} - 1429x^{4} + 25977x^{3} + 395038x^{2} - 9385685x + 43074950$

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
$\operatorname{tr} a_p$	4	16	42	300	−58	792	−400	2738	3174	11244	13748	25426	−14268	−18082	−23084	17522	−36392	27062	37138	−158556	112228	36844	−76350	16100	259432

$\operatorname{tr} a_n$ 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)[1]   # 23.6.a.b

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

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