Atlas › Modular forms › Weight 2 › Level 993 › 993.2.a › 993.2.a.f

Newform orbit 993.2.a.f

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{15} + 3x^{14} - 59x^{13} - 149x^{12} + 1306x^{11} + 2616x^{10} - 14035x^{9} - 19087x^{8} + 81240x^{7} + 50527x^{6} - 249379x^{5} + 22215x^{4} + 314612x^{3} - 215109x^{2} + 17880x + 7342$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
331	$+1$

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	15	−2	17	5	17	1	24	22	16	23	7	−14	16	3	−2	−13	23	40	−12	22	65	−2	−9	30

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.284 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, 3, 331, 993]: [0, 1, 28, 112] (certified at ell = 2147482921)
T = [(2, 1), (5, 1)]; h of degree 55 = 1 x 55 (phi(ord eps) x dim S_k^new); 6 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(993, 2)[5]   # 993.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 993:2 1000