Atlas › Modular forms › Weight 4 › Level 23 › 23.4.a › 23.4.a.a

Newform orbit 23.4.a.a

$q$ -expansion

$q - 2q^{2} - 5q^{3} - 4q^{4} - 6q^{5} + 10q^{6} - 8q^{7} + 24q^{8} - 2q^{9} + 12q^{10} + 34q^{11} + 20q^{12} - 57q^{13} + 16q^{14} + 30q^{15} - 16q^{16} - 80q^{17} + 4q^{18} - 70q^{19} + 24q^{20} + 40q^{21} - 68q^{22} + 23q^{23} - 120q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{3}$ is a root of

$x + 7$

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
$\operatorname{tr} a_p$	−2	−5	−6	−8	34	−57	−80	−70	23	245	103	−298	95	88	−357	−414	−408	822	926	335	−899	−1322	−36	−460	−964

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 23, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.09 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, 7] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 1 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, 4)[0]   # 23.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 23:4 1000