Atlas › Modular forms › Weight 2 › Level 845 › 845.2.a › 845.2.a.a

Newform orbit 845.2.a.a

$q$ -expansion

$q + q^{2} - 2q^{3} - q^{4} + q^{5} - 2q^{6} + 4q^{7} - 3q^{8} + q^{9} + q^{10} - 2q^{11} + 2q^{12} + 4q^{14} - 2q^{15} - q^{16} + 2q^{17} + q^{18} + 6q^{19} - q^{20} - 8q^{21} - 2q^{22} - 6q^{23} + 6q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{3}$ is a root of

$x + 5$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
13	— ( $13^2 \mid 845$ )

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 845.a (Cremona's 845a). Its optimal curve 845a1 is

$y^2 + xy + y = x^3 - 173x + 171$

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	−2	1	4	−2	0	2	6	−6	2	10	2	6	10	−4	2	−6	2	4	−6	6	−12	16	−2	2

$\operatorname{tr} a_n$ 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 845, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.299 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, 5, 13, 65, 169, 845]: [0, 1, 1, 8, 15, 92] (certified at ell = 2147483053)
T = [(2, 1), (3, 3)]; h of degree 51 = 1 x 51 (phi(ord eps) x dim S_k^new); 7 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(845, 2)[0]   # 845.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 845:2 1000