Atlas › Modular forms › Weight 2 › Level 846 › 846.2.a › 846.2.a.h

Newform orbit 846.2.a.h

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 4q^{2} + 4q^{4} - 2q^{5} + 4q^{7} - 4q^{8} + 2q^{10} - 6q^{11} + 6q^{13} - 4q^{14} + 4q^{16} + 8q^{17} + 10q^{19} - 2q^{20} + 6q^{22} - 4q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7} + 5T_{13} + 7T_{17} + 9T_{19}$ , whose minimal polynomial is

$x^{4} - 186x^{3} + 10956x^{2} - 209440x + 730110$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	— ( $3^2 \mid 846$ )
47	$-1$

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
$\operatorname{tr} a_p$	−4	0	−2	4	−6	6	8	10	−4	2	0	8	0	6	4	8	4	0	2	16	0	16	4	40	−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 0.654 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, 3, 6, 9, 18, 47, 94, 141, 282, 423, 846]: [0, 1, 1, 3, 2, 5, 5, 14, 18, 52, 50, 148] (certified at ell = 2147483137)
T = [(5, 1), (7, 3), (13, 5), (17, 7), (19, 9)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 6 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(846, 2)[7]   # 846.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 846:2 1000