Atlas › Modular forms › Weight 2 › Level 910 › 910.2.a › 910.2.a.d

Newform orbit 910.2.a.d

$q$ -expansion

$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + q^{7} - q^{8} - 2q^{9} + q^{10} - 3q^{11} + q^{12} + q^{13} - q^{14} - q^{15} + q^{16} - 6q^{17} + 2q^{18} + 2q^{19} - q^{20} + q^{21} + 3q^{22} - 3q^{23} - q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 3T_{11} + 5T_{17} + 7T_{19} + 9T_{23} + 11T_{29}$ is a root of

$x + 117$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
5	$+1$
7	$-1$
13	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = +1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = -1$ : each L-function in the orbit has odd analytic rank.

Elliptic curve

By modularity this newform is the one attached to the isogeny class 910.d (Cremona's 910b). Its optimal curve 910b1 is

$y^2 + xy + y = x^3 + 6x + 42$

of rank 1 with torsion of order 3. 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	1	−3	1	−6	2	−3	−6	−7	5	9	2	−9	0	0	−13	−13	12	−13	5	6	18	5

$\operatorname{tr} a_n$ 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 910, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.251 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, 5, 7, 10, 13, 14, 26, 35, 65, 70, 91, 130, 182, 455, 910]: [0, 1, 1, 1, 3, 1, 4, 5, 6, 8, 16, 10, 24, 32, 60, 176] (certified at ell = 2147483629)
T = [(3, 1), (11, 3), (17, 5), (19, 7), (23, 9), (29, 11)]; h of degree 25 = 1 x 25 (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(910, 2)[3]   # 910.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 910:2 1000