Atlas › Modular forms › Weight 2 › Level 324 › 324.2.a › 324.2.a.c

Newform orbit 324.2.a.c

$q$ -expansion

$q + 3q^{5} - q^{7} + 3q^{11} - q^{13} + 6q^{17} - 4q^{19} - 3q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13}$ is a root of

$x - 8$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 324$ )
3	— ( $3^2 \mid 324$ )

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 324.c (Cremona's 324a). Its optimal curve 324a1 is

$y^2 = x^3 - 21x + 37$

of rank 0 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	0	3	−1	3	−1	6	−4	−3	3	5	2	3	−1	−9	−6	−3	−13	−7	−12	−10	11	−9	6	11

$\operatorname{tr} a_n$ 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 324, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.234 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, 9, 12, 18, 27, 36, 54, 81, 108, 162, 324]: [0, 1, 1, 2, 3, 2, 5, 5, 4, 9, 11, 10, 21, 29, 57] (certified at ell = 2147483179)
T = [(5, 1), (7, 3), (11, 5), (13, 7)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 1 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(324, 2)[2]   # 324.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 324:2 1000