Atlas › Modular forms › Weight 2 › Level 297 › 297.2.a › 297.2.a.b

Newform orbit 297.2.a.b

$q$ -expansion

$q - q^{2} - q^{4} + 2q^{5} - 5q^{7} + 3q^{8} - 2q^{10} - q^{11} - 2q^{13} + 5q^{14} - q^{16} - 7q^{17} - 2q^{20} + q^{22} + q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 2T_{5} + 3T_{7} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ is a root of

$x + 72$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 297$ )
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 297.b (Cremona's 297b). Its optimal curve 297b1 is

$y^2 + xy + y = x^3 - x^2 + x$

of rank 1 with torsion of order 1. 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
$\operatorname{tr} a_p$	−1	0	2	−5	−1	−2	−7	0	1	−3	−8	−3	11	−9	1	12	−5	6	−4	0	4	5	6	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 297, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.24 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, 3, 9, 11, 27, 33, 99, 297]: [0, 1, 2, 2, 4, 6, 14, 38] (certified at ell = 2147482801)
T = [(2, 1), (5, 2), (7, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 4 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(297, 2)[1]   # 297.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 297:2 1000