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

Newform orbit 843.2.a.f

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q + 5q^{2} - 15q^{3} + 19q^{4} + 7q^{5} - 5q^{6} + 8q^{7} + 18q^{8} + 15q^{9} + 8q^{10} + q^{11} - 19q^{12} + 18q^{13} + 5q^{14} - 7q^{15} + 23q^{16} + 4q^{17} + 5q^{18} + 5q^{19} + 11q^{20} - 8q^{21} + 13q^{22} - 2q^{23} - 18q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{15} - 12x^{14} - 11x^{13} + 606x^{12} - 1229x^{11} - 10897x^{10} + 37345x^{9} + 78441x^{8} - 424377x^{7} - 68941x^{6} + 2137784x^{5} - 1680647x^{4} - 3754271x^{3} + 5655892x^{2} - 1396348x - 335624$

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
$\operatorname{tr} a_p$	5	−15	7	8	1	18	4	5	−2	12	16	54	9	29	−21	42	−22	25	37	−27	33	11	−3	−24	40

$\operatorname{tr} a_n$ 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)[5]   # 843.2.a.f

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

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