Atlas › Modular forms › Weight 2 › Level 850 › 850.2.a › 850.2.a.n

Newform orbit 850.2.a.n

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47}$ , whose minimal polynomial is

$x^{2} + 305x + 16792$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
5	— ( $5^2 \mid 850$ )
17	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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	0	−2	−8	−5	−2	−1	2	1	−9	6	−8	−6	−9	−3	−13	19	10	−21	−7	0	−4	−7	−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 1.428 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, 10, 17, 25, 34, 50, 85, 170, 425, 850]: [0, 1, 1, 3, 2, 3, 6, 9, 10, 30, 46, 138] (certified at ell = 2147482801)
T = [(3, 1), (7, 2), (11, 3), (13, 4), (19, 5), (23, 6), (29, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 8 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(850, 2)[13]   # 850.2.a.n

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

cargo run --release -p sagebrush-web --example atlas -- 850:2 1000