Atlas › Modular forms › Weight 2 › Level 135 › 135.2.a › 135.2.a.a

Newform orbit 135.2.a.a

$q$ -expansion

$q - 2q^{2} + 2q^{4} - q^{5} - 3q^{7} + 2q^{10} - 2q^{11} - 5q^{13} + 6q^{14} - 4q^{16} - 8q^{17} + q^{19} - 2q^{20} + 4q^{22} + 6q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{7}$ is a root of

$x + 11$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 135$ )
5	$+1$

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 135.a (Cremona's 135a). Its optimal curve 135a1 is

$y^2 + y = x^3 - 3x + 4$

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	0	−1	−3	−2	−5	−8	1	6	2	0	5	−10	4	4	−2	−8	7	−9	2	−5	−3	6	−12	−13

$\operatorname{tr} a_n$ 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 135, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.204 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, 5, 9, 15, 27, 45, 135]: [0, 1, 1, 2, 4, 4, 8, 20] (certified at ell = 2147483629)
T = [(2, 1), (7, 3)]; h of degree 6 = 1 x 6 (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(135, 2)[0]   # 135.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 135:2 1000