Atlas › Modular forms › Weight 2 › Level 895 › 895.2.a › 895.2.a.d

Newform orbit 895.2.a.d

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{7} + 7T_{11}$ , whose minimal polynomial is

$x^{8} + 132x^{7} + 6594x^{6} + 147709x^{5} + 1050856x^{4} - 14168047x^{3} - 302711306x^{2} - 1859728751x - 3767597231$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
179	$+1$

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	−8	−2	−16	14	−7	4	−19	−29	−3	−8	−48	5	−2	7	5	−26	38	−28	−9	4	−9	−8	−24

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.429 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, 5, 179, 895]: [0, 1, 16, 92] (certified at ell = 2147483137)
T = [(2, 1), (3, 3), (7, 5), (11, 7)]; h of degree 59 = 1 x 59 (phi(ord eps) x dim S_k^new); 13 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(895, 2)[3]   # 895.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 895:2 1000