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

Newform orbit 910.2.a.o

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{11} + 5T_{17} + 7T_{19} + 9T_{23} + 11T_{29}$ , whose minimal polynomial is

$x^{2} - 97x + 2144$

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 even analytic rank.

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
$\operatorname{tr} a_p$	2	1	2	−2	−1	2	4	6	9	−4	7	−11	−7	−6	5	−6	8	1	13	−4	−21	5	−2	−8	9

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

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)[14]   # 910.2.a.o

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

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