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

Newform orbit 995.2.a.g

Trace form

The 21 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$21q - 3q^{2} + q^{3} + 27q^{4} - 21q^{5} + 3q^{6} + 11q^{7} - 12q^{8} + 30q^{9} + 3q^{10} + 3q^{12} + 33q^{13} + 6q^{14} - q^{15} + 35q^{16} - 15q^{17} - 9q^{18} + 26q^{19} - 27q^{20} + 37q^{21} + 5q^{22} - 8q^{23} + 23q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 21. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{21} + 2x^{20} - 82x^{19} - 149x^{18} + 2605x^{17} + 4484x^{16} - 40805x^{15} - 71145x^{14} + 330596x^{13} + 637823x^{12} - 1301177x^{11} - 3066945x^{10} + 1696858x^{9} + 6688693x^{8} + 1630233x^{7} - 3951835x^{6} - 1606073x^{5} + 841455x^{4} + 365927x^{3} - 57630x^{2} - 19971x + 1247$

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
$\operatorname{tr} a_p$	−3	1	−21	11	0	33	−15	26	−8	12	28	30	18	11	−3	−18	−6	65	17	15	23	2	−2	15	49

$\operatorname{tr} a_n$ 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)[6]   # 995.2.a.g

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

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