Atlas › Modular forms › Weight 6 › Level 85 › 85.6.a › 85.6.a.c

Newform orbit 85.6.a.c

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q - 3q^{2} - 36q^{3} + 187q^{4} - 200q^{5} - 113q^{6} - 184q^{7} - 201q^{8} + 550q^{9} + 75q^{10} - 308q^{11} - 2591q^{12} - 914q^{13} - 3418q^{14} + 900q^{15} + 4307q^{16} - 2312q^{17} - 1024q^{18} - 660q^{19} - 4675q^{20} + 4208q^{21} - 10428q^{22} - 9072q^{23} - 24463q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} + 39x^{7} - 595x^{6} - 26011x^{5} + 89430x^{4} + 2647392x^{3} - 18038204x^{2} + 28736532x + 8865000$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
17	$+1$

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
$\operatorname{tr} a_p$	−3	−36	−200	−184	−308	−914	−2312	−660	−9072	−2362	1464	−19012	−35296	6396	4736	−39090	−53284	−2358	−31532	−117992	12570	122308	−65572	638	−557882

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.255 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, 5, 17, 85]: [1, 3, 8, 46] (certified at ell = 2147483489)
T = [(2, 1), (3, 1)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 5 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(85, 6)[2]   # 85.6.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 85:6 1000