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

Newform orbit 893.2.a.b

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} + 17x^{15} + 93x^{14} + 54x^{13} - 1197x^{12} - 3306x^{11} + 3030x^{10} + 20928x^{9} + 6975x^{8} - 54304x^{7} - 34874x^{6} + 75090x^{5} + 40735x^{4} - 61657x^{3} - 10104x^{2} + 23117x - 5165$

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 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	−13	−1	−9	0	−19	−8	−16	−5	−2	−18	−24	−6	−34	−16	−23	−32	−16	−67	19	−2	−27	−17	20	−50

$\operatorname{tr} a_n$ 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)[1]   # 893.2.a.b

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

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