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

Newform orbit 83.6.a.a

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q - 13q^{2} - 17q^{3} + 133q^{4} - 162q^{5} - 15q^{6} - 247q^{7} - 609q^{8} + 679q^{9} - 519q^{10} - 413q^{11} - 1193q^{12} - 2942q^{13} - 1176q^{14} - 1990q^{15} + 437q^{16} + 69q^{17} - 4532q^{18} - 2446q^{19} - 6353q^{20} - 11006q^{21} - 13799q^{22} - 7823q^{23} - 22851q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{14} + 30x^{13} - 1866x^{12} - 45889x^{11} + 1322678x^{10} + 21384849x^{9} - 479646070x^{8} - 3675520827x^{7} + 88586418009x^{6} + 152523286405x^{5} - 7712884204560x^{4} + 16266519583240x^{3} + 236655664683472x^{2} - 1168728914301808x + 1206108837762336$

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
$\operatorname{tr} a_p$	−13	−17	−162	−247	−413	−2942	69	−2446	−7823	−5449	−9055	−46585	12585	−27158	−12288	−77086	64705	−33501	−125486	57934	−116530	−17628	−96446	−188766	−351786

$\operatorname{tr} a_n$ 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)[0]   # 83.6.a.a

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

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