Atlas › Modular forms › Weight 2 › Level 858 › 858.2.a › 858.2.a.o

Newform orbit 858.2.a.o

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{33})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 2T_{7} + 3T_{23} + 4T_{29} + 5T_{31} + 6T_{17} + 7T_{19} + 8T_{37}$ , whose minimal polynomial is

$x^{2} + 72x + 108$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	$+1$
11	$+1$
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.

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	1	−1	−2	−2	−8	−2	−1	−1	6	−4	5	−11	−8	−20	−17	−1	7	−18	1	−20	−6	16	−20

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.259 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, 6, 11, 13, 22, 26, 33, 39, 66, 78, 143, 286, 429, 858]: [0, 1, 1, 3, 2, 1, 5, 5, 6, 6, 16, 18, 16, 46, 60, 176] (certified at ell = 2147482921)
T = [(5, 1), (7, 2), (23, 3), (29, 4), (31, 5), (17, 6), (19, 7), (37, 8)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 6 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(858, 2)[14]   # 858.2.a.o

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

cargo run --release -p sagebrush-web --example atlas -- 858:2 1000