Atlas › Modular forms › Weight 4 › Level 245 › 245.4.a › 245.4.a.f

Newform orbit 245.4.a.f

$q$ -expansion

$q + 3q^{2} + 2q^{3} + q^{4} - 5q^{5} + 6q^{6} - 21q^{8} - 23q^{9} - 15q^{10} - 45q^{11} + 2q^{12} - 59q^{13} - 10q^{15} - 71q^{16} + 54q^{17} - 69q^{18} + 121q^{19} - 5q^{20} - 135q^{22} + 69q^{23} - 42q^{24} + O(q^{25})$

Coefficient field

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

$x - 9$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
7	— ( $7^2 \mid 245$ )

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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
$\operatorname{tr} a_p$	3	2	−5	0	−45	−59	54	121	69	−162	88	−259	−195	−286	−45	597	360	−392	−280	48	−668	782	−768	1194	−902

$\operatorname{tr} a_n$ 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 245, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.186 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, 5, 7, 35, 49, 245]: [1, 3, 3, 14, 15, 86] (certified at ell = 2147483353)
T = [(2, 1), (3, 3)]; h of degree 41 = 1 x 41 (phi(ord eps) x dim S_k^new); 8 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(245, 4)[5]   # 245.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 245:4 1000