Atlas › Modular forms › Weight 2 › Level 851 › 851.2.a › 851.2.a.g

Newform orbit 851.2.a.g

Trace form

The 20 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$20q + 2q^{2} + 26q^{4} + 13q^{5} + 5q^{6} + 6q^{7} - 3q^{8} + 22q^{9} + q^{10} + 11q^{11} - 8q^{12} + 28q^{13} - 4q^{14} + 5q^{15} + 22q^{16} + 11q^{17} - 5q^{18} + 28q^{19} + 34q^{20} + 29q^{21} - 5q^{22} + 20q^{23} - 14q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 20. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7} + 9T_{11}$ , whose minimal polynomial is

$x^{20} - 208x^{19} + 3388x^{18} + 2132340x^{17} - 151223713x^{16} - 4307020487x^{15} + 781449666982x^{14} - 14616638876400x^{13} - 1370079060656415x^{12} + 65923972828060245x^{11} + 208551791575238647x^{10} - 78038478148583406276x^{9} + 1628425631604100366267x^{8} + 14811156290629794589215x^{7} - 1094965198852843770315877x^{6} + 17104554305115730920860275x^{5} - 87872852528449154733026908x^{4} - 386970594944749584080327831x^{3} + 5205632545788902883224700174x^{2} - 6295679483102233910618200277x - 47130982232540886955321971478$

Atkin–Lehner signs

$p$	$w_p$
23	$-1$
37	$+1$

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
$\operatorname{tr} a_p$	2	0	13	6	11	28	11	28	20	13	19	−20	4	25	1	−3	−24	75	11	−33	−5	33	56	25	33

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.332 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, 23, 37, 851]: [0, 3, 3, 78] (certified at ell = 2147483053)
T = [(2, 1), (3, 3), (5, 5), (7, 7), (11, 9)]; h of degree 67 = 1 x 67 (phi(ord eps) x dim S_k^new); 16 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(851, 2)[6]   # 851.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 851:2 1000