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

Newform orbit 851.2.a.f

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q - q^{2} - 3q^{3} + 5q^{4} - 5q^{5} - 10q^{6} + q^{7} - 9q^{8} + 3q^{9} - 13q^{10} + 2q^{11} + 5q^{12} - 30q^{13} + 2q^{14} - 6q^{15} - q^{16} - 9q^{17} - 7q^{18} - 22q^{19} - 2q^{20} - 28q^{21} - q^{22} - 12q^{23} - 16q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7} + 9T_{11}$ , whose minimal polynomial is

$x^{12} + 10x^{11} - 4362x^{10} - 55243x^{9} + 6388324x^{8} + 91543091x^{7} - 3629687015x^{6} - 52128300110x^{5} + 839282071128x^{4} + 10825494449291x^{3} - 75009481743127x^{2} - 657659522650410x + 2642808494879047$

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 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
$\operatorname{tr} a_p$	−1	−3	−5	1	2	−30	−9	−22	−12	−13	−31	−12	−23	−5	−4	4	−20	−52	15	20	−24	−15	13	−18	−11

$\operatorname{tr} a_n$ 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)[5]   # 851.2.a.f

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

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