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

Newform orbit 845.2.a.b

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{5})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{2} + x - 61$

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
$\operatorname{tr} a_p$	−1	0	−2	−4	−4	0	2	−4	12	−6	0	−6	−6	−8	−8	12	−12	2	−8	8	12	0	0	18	−2

$\operatorname{tr} a_n$ 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)[1]   # 845.2.a.b

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

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