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

Newform orbit 245.4.a.l

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 3q^{2} - 2q^{3} + 13q^{4} - 15q^{5} - 24q^{6} - 15q^{8} + 81q^{9} + 15q^{10} - 74q^{11} + 152q^{12} - 44q^{13} + 10q^{15} - 79q^{16} + 52q^{17} - 411q^{18} - 168q^{19} - 65q^{20} + 184q^{22} - 124q^{23} - 420q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 9x^{2} - 635x + 585$

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	−15	0	−74	−44	52	−168	−124	332	−320	−54	−362	−16	730	110	180	−1222	204	−136	−310	−1034	1660	−242	−100

$\operatorname{tr} a_n$ 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)[11]   # 245.4.a.l

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

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