Atlas › Modular forms › Weight 2 › Level 842 › 842.2.a › 842.2.a.g

Newform orbit 842.2.a.g

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{13} - 13x^{12} + 9x^{11} + 549x^{10} - 2400x^{9} - 1888x^{8} + 30520x^{7} - 46672x^{6} - 57286x^{5} + 165356x^{4} + 14500x^{3} - 201020x^{2} + 20859x + 89892$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
421	$+1$

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
$\operatorname{tr} a_p$	13	6	7	6	−4	22	10	0	−3	7	5	18	15	−2	13	5	−24	12	−4	−19	7	−5	−29	18	6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.26 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, 2, 421, 842]: [0, 1, 35, 107] (certified at ell = 2147482681)
T = [(3, 1), (5, 1)]; h of degree 36 = 1 x 36 (phi(ord eps) x dim S_k^new); 4 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(842, 2)[6]   # 842.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 842:2 1000