Atlas › Modular forms › Weight 2 › Level 702 › 702.2.a › 702.2.a.n

Newform orbit 702.2.a.n

$q$ -expansion

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = 8751T_{5} + 288T_{7} + 4429T_{11} + 5132T_{17} + 663T_{19} + 393T_{23} + 4651T_{29} + 7220T_{31}$ is a root of

$x + 16567$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	— ( $3^2 \mid 702$ )
13	$+1$

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 702.n (Cremona's 702i). Its optimal curve 702i1 is

$y^2 + xy + y = x^3 - x^2 - 29x + 55$

of rank 0 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	1	1	−1	0	0	5	−1	−5	−3	6	1	4	−3	11	4	−8	−8	−16	8	9	−9	−4

$\operatorname{tr} a_n$ 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 702, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.408 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, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702]: [0, 1, 1, 3, 2, 1, 5, 5, 4, 6, 11, 18, 16, 46, 44, 130] (certified at ell = 2147483629)
T = [(5, 8751), (7, 288), (11, 4429), (17, 5132), (19, 663), (23, 393), (29, 4651), (31, 7220)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 10 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(702, 2)[13]   # 702.2.a.n

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

cargo run --release -p sagebrush-web --example atlas -- 702:2 1000