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

Newform orbit 85.6.a.a

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q - 7q^{2} - 36q^{3} + 43q^{4} + 125q^{5} - 105q^{6} - 204q^{7} - 63q^{8} + 531q^{9} - 175q^{10} - 792q^{11} + 785q^{12} + 88q^{13} + 860q^{14} - 900q^{15} - 2365q^{16} + 1445q^{17} - 2052q^{18} - 5160q^{19} + 1075q^{20} - 6428q^{21} - 3058q^{22} - 6140q^{23} - 11751q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{5} + 43x^{4} + 55x^{3} - 9555x^{2} - 24544x + 552560$

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
$\operatorname{tr} a_p$	−7	−36	125	−204	−792	88	1445	−5160	−6140	−7328	−8968	9834	−5302	−4916	−20364	56492	−96200	−7284	−60308	−25304	163328	−50940	−179060	−387356	−106076

$\operatorname{tr} a_n$ 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)[0]   # 85.6.a.a

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

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