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

Newform orbit 245.4.a.n

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + q^{2} + 8q^{3} + 35q^{4} + 25q^{5} - 16q^{6} + 33q^{8} + 81q^{9} + 5q^{10} + 47q^{11} + 98q^{12} - q^{13} + 40q^{15} + 171q^{16} + 2q^{17} - 51q^{18} + 21q^{19} + 175q^{20} + 523q^{22} + 201q^{23} - 848q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{5} - 25x^{4} - 649x^{3} + 12597x^{2} + 80340x - 652400$

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
$\operatorname{tr} a_p$	1	8	25	0	47	−1	2	21	201	190	−388	−145	281	568	473	351	−708	−1944	1118	864	1652	218	1502	−2322	−598

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.186 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)[13]   # 245.4.a.n

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

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