Atlas › Modular forms › Weight 2 › Level 605 › 605.2.a › 605.2.a.d

Newform orbit 605.2.a.d

Trace form

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

$2q - 2q^{2} + 2q^{4} - 2q^{5} + 8q^{6} + 4q^{7} - 6q^{8} + 10q^{9} + 2q^{10} - 16q^{12} + 8q^{13} - 4q^{14} + 6q^{16} - 8q^{17} - 10q^{18} - 2q^{20} + 8q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{2})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{7} + 7T_{13}$ , whose minimal polynomial is

$x^{2} - 74x + 487$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
11	— ( $11^2 \mid 605$ )

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	0	−2	4	0	8	−8	0	0	−4	0	−4	−12	12	0	12	−8	−4	8	0	8	−8	12	−4	−4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.268 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, 11, 55, 121, 605]: [0, 1, 2, 8, 12, 68] (certified at ell = 2147482921)
T = [(2, 1), (3, 3), (7, 5), (13, 7)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 9 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(605, 2)[3]   # 605.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 605:2 1000