Atlas › Modular forms › Weight 2 › Level 924 › 924.2.a › 924.2.a.f

Newform orbit 924.2.a.f

$q$ -expansion

$q + q^{3} - 3q^{5} + q^{7} + q^{9} + q^{11} - 7q^{13} - 3q^{15} - 6q^{17} - q^{19} + q^{21} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 2T_{13} + 3T_{17} + 4T_{19} + 5T_{23} + 6T_{29} + 7T_{31} + 8T_{37} + 9T_{41} + 10T_{43} + 11T_{47} + 12T_{53} + 13T_{59} + 14T_{61} + 15T_{67} + 16T_{71}$ is a root of

$x + 774$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 924$ )
3	$-1$
7	$-1$
11	$-1$

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 924.f (Cremona's 924h). Its optimal curve 924h1 is

$y^2 = x^3 + x^2 - 17242x + 875009$

of rank 1 with torsion of order 3. 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	1	−3	1	1	−7	−6	−1	0	−3	2	5	−12	8	−3	−12	−3	−10	−13	−12	11	8	6	6	8

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

Sato–Tate

The normalized $a_p / 2p^{1/2}$ for the primes $p < 1000$ not dividing 924, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.954 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, 7, 11, 12, 14, 21, 22, 28, 33, 42, 44, 66, 77, 84, 132, 154, 231, 308, 462, 924]: [0, 1, 1, 2, 3, 1, 2, 5, 4, 4, 5, 7, 6, 12, 9, 16, 10, 22, 30, 28, 36, 54, 104, 204] (certified at ell = 2147482951)
T = [(5, 1), (13, 2), (17, 3), (19, 4), (23, 5), (29, 6), (31, 7), (37, 8), (41, 9), (43, 10), (47, 11), (53, 12), (59, 13), (61, 14), (67, 15), (71, 16)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 3 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(924, 2)[5]   # 924.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 924:2 1000