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

Newform orbit 245.4.a.h

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 6q^{2} + 2q^{3} + 6q^{4} - 10q^{5} + 6q^{6} + 6q^{8} - 16q^{9} + 30q^{10} + 28q^{11} - 66q^{12} + 36q^{13} - 10q^{15} - 14q^{16} + 76q^{17} + 72q^{18} - 160q^{19} - 30q^{20} - 44q^{22} - 22q^{23} + 162q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{2})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{2} - 200$

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
$\operatorname{tr} a_p$	−6	2	−10	0	28	36	76	−160	−22	−250	132	−416	106	−666	196	−952	−840	−98	−1286	1064	172	−1240	1906	−650	628

$\operatorname{tr} a_n$ 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)[7]   # 245.4.a.h

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

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