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

Newform orbit 845.2.a.f

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{13})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{2} - 7x + 9$

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	2	2	2	4	0	8	4	−6	2	8	0	−6	−6	−4	8	0	−2	14	12	16	−4	4	−2	24

$\operatorname{tr} a_n$ 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)[5]   # 845.2.a.f

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

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