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

Newform orbit 895.2.a.f

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$18q + 7q^{2} - 3q^{3} + 23q^{4} - 18q^{5} + 13q^{6} + 18q^{8} + 33q^{9} - 7q^{10} + 16q^{11} + 5q^{12} - 22q^{13} + 8q^{14} + 3q^{15} + 37q^{16} + 9q^{17} + 26q^{18} + 4q^{19} - 23q^{20} + 19q^{21} + 7q^{22} + 15q^{23} + 8q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{7} + 7T_{11}$ , whose minimal polynomial is

$x^{18} - 110x^{17} - 4213x^{16} + 918833x^{15} - 16434072x^{14} - 2164857150x^{13} + 84726528918x^{12} + 1807078326562x^{11} - 132319155799468x^{10} + 113870911349733x^{9} + 96018431523887336x^{8} - 1061750455780271693x^{7} - 32531756484656953766x^{6} + 617840586698701060528x^{5} + 3495377685071056841172x^{4} - 129778599023917311122306x^{3} + 358731261604465608348899x^{2} + 6636977606272453860234519x - 33840233262798879688595795$

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 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
$\operatorname{tr} a_p$	7	−3	−18	0	16	−22	9	4	15	25	−3	−24	56	−15	10	−1	9	16	10	64	−19	22	−37	22	2

$\operatorname{tr} a_n$ 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)[5]   # 895.2.a.f

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

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