Atlas › Modular forms › Weight 2 › Level 870 › 870.2.a › 870.2.a.k

Newform orbit 870.2.a.k

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{2})$ . It is generated by the eigenvalue $\alpha$ of $T = 9399T_{7} + 2198T_{11} + 5531T_{19} + 9022T_{23} + 6143T_{31} + 7295T_{13} + 377T_{17} + 2724T_{37}$ , whose minimal polynomial is

$x^{2} - 74996x + 1387641116$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	$+1$
5	$-1$
29	$+1$

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	−2	2	4	−4	8	4	0	0	−2	−4	4	−8	8	0	20	0	8	16	0	28	4	4	8	−12

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.322 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, 3, 5, 6, 10, 15, 29, 30, 58, 87, 145, 174, 290, 435, 870]: [0, 1, 1, 1, 3, 3, 4, 3, 10, 9, 12, 16, 34, 48, 64, 188] (certified at ell = 2147483549)
T = [(7, 9399), (11, 2198), (19, 5531), (23, 9022), (31, 6143), (13, 7295), (17, 377), (37, 2724)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 11 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(870, 2)[10]   # 870.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 870:2 1000