Atlas › Modular forms › Weight 2 › Level 845 › 845.2.a › 845.2.a.n

Newform orbit 845.2.a.n

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q - 3q^{2} + 7q^{3} + 17q^{4} - 9q^{5} + 2q^{6} - 7q^{7} - 12q^{8} + 16q^{9} + 3q^{10} + 9q^{11} + 12q^{12} - 2q^{14} - 7q^{15} + 37q^{16} - q^{17} + 10q^{18} + 4q^{19} - 17q^{20} + q^{21} + 12q^{22} + 14q^{23} + 35q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{9} - 18x^{8} - 55x^{7} + 2279x^{6} - 4679x^{5} - 77988x^{4} + 282570x^{3} + 444367x^{2} - 1591177x - 1195649$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
13	— ( $13^2 \mid 845$ )

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	7	−9	−7	9	0	−1	4	14	12	7	5	10	39	−36	−8	21	−3	−1	1	0	39	−7	19	34

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.299 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, 5, 13, 65, 169, 845]: [0, 1, 1, 8, 15, 92] (certified at ell = 2147483053)
T = [(2, 1), (3, 3)]; h of degree 51 = 1 x 51 (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(845, 2)[13]   # 845.2.a.n

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

cargo run --release -p sagebrush-web --example atlas -- 845:2 1000