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

Newform orbit 83.4.a.a

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q - 5q^{2} - 11q^{3} + 13q^{4} - 17q^{5} - 27q^{6} - 46q^{7} - 57q^{8} + 16q^{9} - 65q^{10} - 59q^{11} - 89q^{12} - 205q^{13} + 30q^{14} - 136q^{15} - 163q^{16} - 148q^{17} - 188q^{18} - 29q^{19} + 3q^{20} - 197q^{21} - 127q^{22} + 159q^{23} + 465q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{7} + 16x^{6} + 18x^{5} - 406x^{4} - 1175x^{3} + 1778x^{2} + 9644x + 8964$

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 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
$\operatorname{tr} a_p$	−5	−11	−17	−46	−59	−205	−148	−29	159	29	−420	−1091	127	−49	150	−785	−81	−195	−1825	−666	−1594	−968	581	1616	50

$\operatorname{tr} a_n$ 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)[0]   # 83.4.a.a

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

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