Atlas › Modular forms › Weight 4 › Level 230 › 230.4.a › 230.4.a.a

Newform orbit 230.4.a.a

$q$ -expansion

$q - 2q^{2} - 5q^{3} + 4q^{4} - 5q^{5} + 10q^{6} + 12q^{7} - 8q^{8} - 2q^{9} + 10q^{10} + 22q^{11} - 20q^{12} + 19q^{13} - 24q^{14} + 25q^{15} + 16q^{16} + 96q^{17} + 4q^{18} - 98q^{19} - 20q^{20} - 60q^{21} - 44q^{22} + 23q^{23} + 40q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 3T_{7}$ is a root of

$x - 31$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
5	$+1$
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	−5	12	22	19	96	−98	23	−227	−285	−398	271	−100	−285	18	−352	−478	330	835	−1127	322	572	−504	1712

$\operatorname{tr} a_n$ 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 230, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.206 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, 2, 5, 10, 23, 46, 115, 230]: [1, 2, 3, 7, 7, 20, 38, 112] (certified at ell = 2147483053)
T = [(3, 1), (7, 3)]; h of degree 22 = 1 x 22 (phi(ord eps) x dim S_k^new); 6 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(230, 4)[0]   # 230.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 230:4 1000