Atlas › Modular forms › Weight 2 › Level 708 › 708.2.a › 708.2.a.d

Newform orbit 708.2.a.d

Trace form

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

$2q + 2q^{3} - 4q^{5} - q^{7} + 2q^{9} - 10q^{11} - 8q^{13} - 4q^{15} - 9q^{17} - q^{19} - q^{21} + q^{23} + O(q^{25})$

Coefficient field

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

$x^{2} + 5x + 5$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 708$ )
3	$-1$
59	$+1$

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
$\operatorname{tr} a_p$	0	2	−4	−1	−10	−8	−9	−1	1	5	7	−5	−13	−10	7	4	−2	1	2	0	−7	12	−11	−11	−6

$\operatorname{tr} a_n$ 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, 2, 3, 4, 6, 12, 59, 118, 177, 236, 354, 708]: [0, 1, 1, 2, 3, 5, 6, 17, 22, 33, 64, 126] (certified at ell = 2147482481)
T = [(5, 1), (7, 1)]; h of degree 10 = 1 x 10 (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(708, 2)[3]   # 708.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 708:2 1000