Atlas › Modular forms › Weight 2 › Level 862 › 862.2.a › 862.2.a.h

Newform orbit 862.2.a.h

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11} + 9T_{13}$ , whose minimal polynomial is

$x^{5} - 51x^{4} - 535x^{3} + 49875x^{2} - 132866x - 9498413$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
431	$+1$

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	5	3	−3	1	5	7	2	19	−18	16	−7	−1	3	−6	12	10	2	−6	−14	−15	16	−4	−21	−20

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.47 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, 431, 862]: [0, 1, 37, 110] (certified at ell = 2147481491)
T = [(3, 1), (5, 3), (7, 5), (11, 7), (13, 9)]; h of degree 35 = 1 x 35 (phi(ord eps) x dim S_k^new); 9 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(862, 2)[7]   # 862.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 862:2 1000