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

Newform orbit 893.2.a.a

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{12} + 10x^{11} + 11x^{10} - 155x^{9} - 367x^{8} + 677x^{7} + 1892x^{6} - 973x^{5} - 2235x^{4} + 632x^{3} + 588x^{2} - 117x - 27$

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
$\operatorname{tr} a_p$	−1	−9	−7	−13	−4	−17	−6	12	−13	−2	−14	−2	8	−42	12	3	8	−6	−29	−7	−48	−11	−57	−2	−14

$\operatorname{tr} a_n$ 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)[0]   # 893.2.a.a

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

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