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

Newform orbit 245.4.a.p

Trace form

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

$6q - 2q^{2} + 16q^{3} + 14q^{4} - 30q^{5} + 24q^{6} - 66q^{8} + 70q^{9} + 10q^{10} - 16q^{11} + 160q^{12} + 168q^{13} - 80q^{15} + 298q^{16} - 4q^{17} + 354q^{18} + 308q^{19} - 70q^{20} - 236q^{22} - 336q^{23} - 92q^{24} + O(q^{25})$

Coefficient field

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

$x^{6} - 46x^{5} - 89x^{4} + 16556x^{3} + 37007x^{2} - 847710x + 536609$

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
$\operatorname{tr} a_p$	−2	16	−30	0	−16	168	−4	308	−336	176	392	−140	656	−388	628	−676	996	740	1768	−224	2640	1636	140	−1904	516

$\operatorname{tr} a_n$ 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)[15]   # 245.4.a.p

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

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