Atlas › Modular forms › Weight 6 › Level 83 › 83.6.a › 83.6.a.b

Newform orbit 83.6.a.b

Trace form

The 20 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$20q + 7q^{2} + 37q^{3} + 373q^{4} + 88q^{5} + 129q^{6} + 439q^{7} + 351q^{8} + 2137q^{9} + 481q^{10} + 313q^{11} + 1399q^{12} + 2804q^{13} + 1610q^{15} + 8885q^{16} + 2381q^{17} + 3568q^{18} + 2608q^{19} + 1647q^{20} + 5881q^{21} + 18703q^{22} + 14802q^{23} + 34687q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 20. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{20} - 44x^{19} - 3166x^{18} + 147918x^{17} + 4167415x^{16} - 207251872x^{15} - 3004775505x^{14} + 157725819186x^{13} + 1321448502777x^{12} - 71055482224630x^{11} - 376148208590796x^{10} + 19333515828475298x^{9} + 71975712093366637x^{8} - 3099869138360500064x^{7} - 9289698811777097141x^{6} + 269978724552071367110x^{5} + 742441695991257810002x^{4} - 10241142817009646582134x^{3} - 27466260911857785565328x^{2} + 52924821601381361181808x + 24826081189085403897680$

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
$\operatorname{tr} a_p$	7	37	88	439	313	2804	2381	2608	14802	19515	13135	45999	−4772	16018	−9788	21376	40943	27191	64688	63938	65234	−64074	137780	−88378	312918

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.249 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, 36] (certified at ell = 2147483323)
T = [(2, 1), (3, 1)]; h of degree 34 = 1 x 34 (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(83, 6)[1]   # 83.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 83:6 1000