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

Newform orbit 843.2.a.d

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q - 3q^{2} - 8q^{3} + 5q^{4} - 3q^{5} + 3q^{6} - 9q^{7} + 8q^{9} - 6q^{10} + 10q^{11} - 5q^{12} - 12q^{13} + 3q^{15} - 9q^{16} - 6q^{17} - 3q^{18} - 3q^{19} + 3q^{20} + 9q^{21} - 10q^{22} - 9q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{8} + 6x^{7} - x^{6} - 30x^{5} + 4x^{4} + 29x^{3} + 2x^{2} - 5x - 1$

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 odd 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
$\operatorname{tr} a_p$	−3	−8	−3	−9	10	−12	−6	−3	−9	−2	−5	−38	4	−18	18	−36	18	−9	−31	20	−21	−6	12	11	−30

$\operatorname{tr} a_n$ 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)[3]   # 843.2.a.d

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

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