Atlas › Modular forms › Weight 2 › Level 245 › 245.2.a › 245.2.a.d

Newform orbit 245.2.a.d

Trace form

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

$2q - q^{2} + q^{3} + 5q^{4} - 2q^{5} + 8q^{6} - 9q^{8} + 3q^{9} + q^{10} + q^{11} - 6q^{12} - 5q^{13} - q^{15} + 3q^{16} + 5q^{17} + 7q^{18} + 6q^{19} - 5q^{20} + 8q^{22} - 2q^{23} + 4q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{2} - 2x - 67$

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	1	−2	0	1	−5	5	6	−2	1	0	12	−2	10	5	−2	8	−6	4	16	8	−9	−8	−6	9

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.194 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 5, 7, 35, 49, 245]: [0, 1, 1, 6, 5, 30] (certified at ell = 2147483353)
T = [(2, 1), (3, 3)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 2 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, 2)[3]   # 245.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 245:2 1000