Atlas › Modular forms › Weight 2 › Level 995 › 995.2.a › 995.2.a.e

Newform orbit 995.2.a.e

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{13} - 28x^{11} + 13x^{10} + 281x^{9} - 268x^{8} - 1127x^{7} + 1711x^{6} + 1053x^{5} - 3225x^{4} + 1605x^{3} + 364x^{2} - 475x + 97$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
199	$+1$

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	−1	−13	−5	−12	−23	11	−26	12	−20	−16	−28	−30	−5	−7	6	−28	−51	−29	−3	−23	−10	8	−33	−13

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.376 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, 199, 995]: [0, 1, 17, 102] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 67 = 1 x 67 (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(995, 2)[4]   # 995.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 995:2 1000