Atlas › Modular forms › Weight 2 › Level 893 › 893.2.a › 893.2.a.c

Newform orbit 893.2.a.c

Trace form

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

$18q + q^{2} + 13q^{3} + 21q^{4} + 5q^{5} + 6q^{6} + 3q^{7} + 6q^{8} + 19q^{9} + 7q^{10} + 6q^{11} + 34q^{12} + 21q^{13} - 9q^{14} + 19q^{15} + 23q^{16} - 4q^{17} - 19q^{18} - 18q^{19} + 9q^{20} + q^{21} + 28q^{22} + 9q^{23} + 6q^{24} + O(q^{25})$

Coefficient field

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

$x^{18} - 14x^{17} + 27x^{16} + 449x^{15} - 2194x^{14} - 2501x^{13} + 31566x^{12} - 20376x^{11} - 187457x^{10} + 236402x^{9} + 568307x^{8} - 838197x^{7} - 1011786x^{6} + 1362149x^{5} + 1123987x^{4} - 1004995x^{3} - 703855x^{2} + 249097x + 163363$

Atkin–Lehner signs

$p$	$w_p$
19	$+1$
47	$-1$

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
$\operatorname{tr} a_p$	1	13	5	3	6	21	−4	−18	9	12	26	8	12	28	18	5	30	−16	45	−1	−24	33	−13	−6	44

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.27 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, 19, 47, 893]: [0, 2, 5, 82] (certified at ell = 2147479273)
T = [(2, 1), (3, 1)]; h of degree 69 = 1 x 69 (phi(ord eps) x dim S_k^new); 7 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(893, 2)[2]   # 893.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 893:2 1000