Atlas › Modular forms › Weight 2 › Level 918 › 918.2.a › 918.2.a.h

Newform orbit 918.2.a.h

$q$ -expansion

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = 3208T_{5} + 171T_{7} + 940T_{11} + 6115T_{13} + 4440T_{19} + 6039T_{23} + 7794T_{29} + 3537T_{31}$ is a root of

$x - 2077$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	— ( $3^2 \mid 918$ )
17	$-1$

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 918.h (Cremona's 918h). Its optimal curve 918h1 is

$y^2 + xy + y = x^3 - x^2 - 86x + 357$

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	−2	0	−6	1	−8	−3	10	6	−7	−11	2	−10	11	3	2	−6	0	−4	10	3	−4	0

$\operatorname{tr} a_n$ 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 918, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.915 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, 6, 9, 17, 18, 27, 34, 51, 54, 102, 153, 306, 459, 918]: [0, 1, 1, 3, 2, 2, 5, 4, 6, 8, 11, 22, 20, 58, 56, 166] (certified at ell = 2147482801)
T = [(5, 3208), (7, 171), (11, 940), (13, 6115), (19, 4440), (23, 6039), (29, 7794), (31, 3537)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 12 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(918, 2)[7]   # 918.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 918:2 1000