Atlas › Modular forms › Weight 2 › Level 90 › 90.2.a › 90.2.a.b

Newform orbit 90.2.a.b

$q$ -expansion

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{7} + 3T_{11} + 5T_{13} + 7T_{17}$ is a root of

$x - 6$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	— ( $3^2 \mid 90$ )
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 90.b (Cremona's 90b). Its optimal curve 90b1 is

$y^2 + xy + y = x^3 - x^2 - 8x + 11$

of rank 0 with torsion of order 6. 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	−1	2	−6	−4	6	−4	0	6	−4	8	0	8	0	6	−6	2	−4	12	−10	−4	−12	−12	2

$\operatorname{tr} a_n$ 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 90, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.103 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, 5, 6, 9, 10, 15, 18, 30, 45, 90]: [0, 1, 1, 1, 3, 2, 3, 4, 5, 10, 8, 22] (certified at ell = 2147483629)
T = [(7, 1), (11, 3), (13, 5), (17, 7)]; h of degree 3 = 1 x 3 (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(90, 2)[1]   # 90.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 90:2 1000