Atlas › Modular forms › Weight 2 › Level 735 › 735.2.a › 735.2.a.f

Newform orbit 735.2.a.f

$q$ -expansion

$q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} - 3q^{8} + q^{9} - q^{10} + q^{12} + 6q^{13} + q^{15} - q^{16} - 2q^{17} + q^{18} + 8q^{19} + q^{20} + 8q^{23} + 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{11} + 5T_{13} + 7T_{17} + 9T_{19}$ is a root of

$x - 89$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
5	$+1$
7	— ( $7^2 \mid 735$ )

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 735.f (Cremona's 735a). Its optimal curve 735a1 is

$y^2 + xy = x^3 + x^2 - 123x - 552$

of rank 0 with torsion of order 2. 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	−1	−1	0	0	6	−2	8	8	−2	−4	−2	6	4	−8	10	−4	2	4	−12	2	8	4	6	18

$\operatorname{tr} a_n$ 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 735, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.693 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, 7, 15, 21, 35, 49, 105, 147, 245, 735]: [0, 1, 1, 1, 4, 4, 6, 5, 20, 20, 30, 116] (certified at ell = 2147483353)
T = [(2, 1), (11, 3), (13, 5), (17, 7), (19, 9)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 8 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(735, 2)[5]   # 735.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 735:2 1000