Atlas › Modular forms › Weight 2 › Level 775 › 775.2.a › 775.2.a.b

Newform orbit 775.2.a.b

$q$ -expansion

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 2T_{3} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{17} + 7T_{19} + 8T_{23}$ is a root of

$x - 61$

Atkin–Lehner signs

$p$	$w_p$
5	— ( $5^2 \mid 775$ )
31	$-1$

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 775.b (Cremona's 775b). Its optimal curve 775b1 is

$y^2 + xy + y = x^3 - 26x - 177$

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	0	−4	4	0	8	4	−2	−6	1	4	−6	6	−8	12	−4	10	−8	0	4	0	−2	14	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 775, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.451 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, 25, 31, 155, 775]: [0, 1, 3, 3, 18, 82] (certified at ell = 2147482921)
T = [(2, 1), (3, 2), (7, 3), (11, 4), (13, 5), (17, 6), (19, 7), (23, 8)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 13 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(775, 2)[1]   # 775.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 775:2 1000