Atlas › Modular forms › Weight 2 › Level 748 › 748.2.a › 748.2.a.a

Newform orbit 748.2.a.a

$q$ -expansion

$q + 3q^{3} + 3q^{5} - 2q^{7} + 6q^{9} - q^{11} - 2q^{13} + 9q^{15} - q^{17} + 2q^{19} - 6q^{21} + 3q^{23} + O(q^{25})$

Coefficient field

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

$x - 6$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 748$ )
11	$+1$
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 748.a (Cremona's 748a). Its optimal curve 748a1 is

$y^2 = x^3 - 496x - 4252$

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
$\operatorname{tr} a_p$	0	3	3	−2	−1	−2	−1	2	3	−10	7	−7	4	2	0	−10	−5	0	−9	−7	14	4	18	−9	−17

$\operatorname{tr} a_n$ 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 748, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.328 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, 4, 11, 17, 22, 34, 44, 68, 187, 374, 748]: [0, 1, 2, 2, 2, 5, 6, 9, 12, 20, 58, 114] (certified at ell = 2147482801)
T = [(3, 1), (5, 3), (7, 5), (13, 7), (19, 9)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 3 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(748, 2)[0]   # 748.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 748:2 1000