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

Newform orbit 995.2.a.f

Trace form

The 19 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$19q + 5q^{2} + 7q^{3} + 25q^{4} + 19q^{5} + 3q^{6} + 7q^{7} + 18q^{8} + 16q^{9} + 5q^{10} - 3q^{12} + 37q^{13} - 18q^{14} + 7q^{15} + 29q^{16} + 23q^{17} + 7q^{18} + 14q^{19} + 25q^{20} + 21q^{21} + 7q^{22} + 8q^{23} - 21q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 19. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{19} - 12x^{18} + x^{17} + 503x^{16} - 1520x^{15} - 6305x^{14} + 33353x^{13} + 14028x^{12} - 279357x^{11} + 238371x^{10} + 967936x^{9} - 1613216x^{8} - 867930x^{7} + 3045679x^{6} - 1125219x^{5} - 1000302x^{4} + 756078x^{3} - 129429x^{2} - 4959x + 1611$

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 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
$\operatorname{tr} a_p$	5	7	19	7	0	37	23	14	8	−4	8	24	−6	−1	13	20	−12	25	17	−27	47	2	32	15	31

$\operatorname{tr} a_n$ 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)[5]   # 995.2.a.f

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

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