Atlas › Modular forms › Weight 2 › Level 800 › 800.2.a › 800.2.a.n

Newform orbit 800.2.a.n

Trace form

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

$2q + 2q^{3} + 6q^{7} + 6q^{9} - 4q^{21} + 2q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{5})$ . It is generated by the eigenvalue $\alpha$ of $T = 3990T_{3} + 2855T_{7} + 5427T_{11} + 9524T_{13} + 7538T_{17} + 6229T_{19} + 7256T_{23} + 6862T_{29}$ , whose minimal polynomial is

$x^{2} + 42722x - 1672311124$

Complex multiplication by $\mathbb{Q}(\sqrt{-20})$ : $a_p = 0$ for every conjugate at the 5 primes $p \le 31$ inert in it (exactly, from $\operatorname{tr} a_{p^2} = -\dim \cdot p^{k-1}$ ), and $\operatorname{tr} a_p = 0$ at all 85 inert primes $p < 1000$ . This is evidence from the stored coefficients, not a proof.

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 800$ )
5	— ( $5^2 \mid 800$ )

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	0	6	0	0	0	0	2	−12	0	0	0	18	14	0	0	0	6	0	0	0	−22	12	0

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.312 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, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 80, 100, 160, 200, 400, 800]: [0, 1, 2, 1, 3, 3, 4, 6, 3, 6, 10, 9, 16, 18, 28, 34, 64, 124] (certified at ell = 2147482921)
T = [(3, 3990), (7, 2855), (11, 5427), (13, 9524), (17, 7538), (19, 6229), (23, 7256), (29, 6862)]; h of degree 19 = 1 x 19 (phi(ord eps) x dim S_k^new); 12 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(800, 2)[13]   # 800.2.a.n

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

cargo run --release -p sagebrush-web --example atlas -- 800:2 1000