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

Newform orbit 850.2.a.p

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. 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

$x^{3} - 457x^{2} + 65424x - 2914400$

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
$\operatorname{tr} a_p$	−3	−1	0	0	6	−1	−3	1	12	17	3	−8	16	−4	15	5	9	−9	14	−1	−1	4	20	−9	−3

$\operatorname{tr} a_n$ 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)[15]   # 850.2.a.p

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

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