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

Newform orbit 842.2.a.e

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{4} + 7x^{3} + 14x^{2} + 4x - 7$

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 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
$\operatorname{tr} a_p$	4	−3	−4	−9	−1	−16	−11	4	1	−10	−6	8	−13	−6	3	−9	4	8	−11	−11	−13	16	14	−6	1

$\operatorname{tr} a_n$ 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)[4]   # 842.2.a.e

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

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