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

Newform orbit 800.2.a.d

$q$ -expansion

$q - 3q^{9} - 6q^{13} - 2q^{17} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = 3990T_{3} + 2855T_{7} + 5427T_{11} + 9524T_{13} + 7538T_{17} + 6229T_{19} + 7256T_{23} + 6862T_{29}$ is a root of

$x + 140840$

Complex multiplication by $\mathbb{Q}(\sqrt{-4})$ : $a_p = 0$ for every conjugate at the 6 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 87 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 $a_p = -p^{k/2-1} w_p$ .

Elliptic curve

By modularity this newform is the one attached to the isogeny class 800.d (Cremona's 800a). Its optimal curve 800a1 is

$y^2 = x^3 - 25x$

of rank 1 with torsion of order 4. Its $a_p$ , computed by Sagebrush's point-counting engine, equal the newform's for every prime $p < 1000$ .

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	0	0	0	0	−6	−2	0	0	−10	0	2	10	0	0	−14	0	−10	0	0	6	0	0	10	−18

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

Sato–Tate

With CM the normalized $a_p / 2p^{1/2}$ for the primes $p < 1000$ not dividing 800, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ (CM forms follow a different law):

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)[3]   # 800.2.a.d

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

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