Atlas › Modular forms › Weight 4 › Level 83 › 83.4.a › 83.4.a.b

Newform orbit 83.4.a.b

Trace form

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

$13q + 5q^{2} + 7q^{3} + 73q^{4} + 33q^{5} - 3q^{6} + 52q^{7} + 63q^{8} + 178q^{9} + 35q^{10} + 7q^{11} + 127q^{12} + 237q^{13} + 114q^{14} + 104q^{15} + 365q^{16} - 12q^{17} + 262q^{18} + 237q^{19} + 403q^{20} + 121q^{21} - 121q^{22} - 59q^{23} - 1253q^{24} + O(q^{25})$

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

$x^{13} - 12x^{12} - 278x^{11} + 3262x^{10} + 27544x^{9} - 320612x^{8} - 1210811x^{7} + 14512388x^{6} + 21941285x^{5} - 308390534x^{4} - 49861687x^{3} + 2605953954x^{2} - 1837881750x - 3474959174$

Atkin–Lehner signs

$p$	$w_p$
83	$+1$

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	33	52	7	237	−12	237	−59	−255	−154	1169	−351	1013	514	1505	501	1649	2277	798	2042	808	−1079	1692	4274

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.1 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 engine's checks

dim M^+ at levels [1, 83]: [1, 22] (certified at ell = 2147483323)
T = [(2, 1), (3, 1)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 3 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(83, 4)[1]   # 83.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 83:4 1000