Atlas › Modular forms › Weight 2 › Level 712 › 712.2.a › 712.2.a.a

Newform orbit 712.2.a.a

$q$ -expansion

$q - 2q^{3} + 2q^{5} - 4q^{7} + q^{9} + 4q^{13} - 4q^{15} + 2q^{17} + 2q^{19} + 8q^{21} + 4q^{23} + O(q^{25})$

Coefficient field

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

$x - 4$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 712$ )
89	$-1$

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 712.a (Cremona's 712a). Its optimal curve 712a1 is

$y^2 = x^3 + x^2 - 32x - 80$

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
$\operatorname{tr} a_p$	0	−2	2	−4	0	4	2	2	4	−8	4	8	6	10	8	−2	−2	4	−8	8	−2	8	10	1	14

$\operatorname{tr} a_n$ 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 712, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.249 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, 8, 89, 178, 356, 712]: [0, 1, 2, 3, 8, 24, 48, 94] (certified at ell = 2147482921)
T = [(3, 1), (5, 3)]; h of degree 22 = 1 x 22 (phi(ord eps) x dim S_k^new); 4 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(712, 2)[0]   # 712.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 712:2 1000