Atlas › Modular forms › Weight 2 › Level 245 › 245.2.a › 245.2.a.a

Newform orbit 245.2.a.a

$q$ -expansion

$q - 2q^{2} - 3q^{3} + 2q^{4} + q^{5} + 6q^{6} + 6q^{9} - 2q^{10} + q^{11} - 6q^{12} - 3q^{13} - 3q^{15} - 4q^{16} + 3q^{17} - 12q^{18} - 6q^{19} + 2q^{20} - 2q^{22} - 4q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{3}$ is a root of

$x + 11$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
7	— ( $7^2 \mid 245$ )

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 245.a (Cremona's 245a). Its optimal curve 245a1 is

$y^2 + y = x^3 - 7x + 12$

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
$\operatorname{tr} a_p$	−2	−3	1	0	1	−3	3	−6	−4	−1	−6	0	−6	−6	9	−10	6	0	−14	−8	−6	−1	−12	−12	15

$\operatorname{tr} a_n$ 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 245, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.194 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, 5, 7, 35, 49, 245]: [0, 1, 1, 6, 5, 30] (certified at ell = 2147483353)
T = [(2, 1), (3, 3)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 2 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(245, 2)[0]   # 245.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 245:2 1000