Atlas › Modular forms › Weight 2 › Level 843 › 843.2.a › 843.2.a.e

Newform orbit 843.2.a.e

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q + 6q^{2} + 14q^{3} + 16q^{4} + 13q^{5} + 6q^{6} + 15q^{8} + 14q^{9} + 2q^{10} + 13q^{11} + 16q^{12} + 8q^{13} + q^{14} + 13q^{15} + 16q^{16} + 8q^{17} + 6q^{18} - 9q^{19} + 17q^{20} - 3q^{22} + 18q^{23} + 15q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{14} - 19x^{13} + 112x^{12} + 2x^{11} - 2413x^{10} + 7412x^{9} + 6223x^{8} - 65750x^{7} + 83567x^{6} + 93244x^{5} - 282508x^{4} + 147619x^{3} + 81776x^{2} - 75708x + 7160$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
281	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

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
$\operatorname{tr} a_p$	6	14	13	0	13	8	8	−9	18	20	−8	28	13	−11	33	28	26	−17	−5	25	1	−9	17	32	−12

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.35 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, 3, 281, 843]: [0, 1, 24, 96] (certified at ell = 2147482681)
T = [(2, 1), (5, 1)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 5 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(843, 2)[4]   # 843.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 843:2 1000