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

Newform orbit 85.6.a.d

Trace form

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

$8q + 5q^{2} + 18q^{3} + 187q^{4} + 200q^{5} + 111q^{6} - 46q^{7} + 183q^{8} + 1066q^{9} + 125q^{10} + 1386q^{11} - 709q^{12} + 354q^{13} - 60q^{14} + 450q^{15} + 3011q^{16} - 2312q^{17} + 1698q^{18} + 784q^{19} + 4675q^{20} + 8236q^{21} + 6430q^{22} + 5830q^{23} + 9945q^{24} + O(q^{25})$

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

$x^{8} - 23x^{7} - 1573x^{6} + 32607x^{5} + 751032x^{4} - 11660860x^{3} - 125115228x^{2} + 599357148x + 5114156640$

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 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
$\operatorname{tr} a_p$	5	18	200	−46	1386	354	−2312	784	5830	4982	15074	1672	10220	1808	3240	−31950	59232	−106458	−32244	46254	−15426	275378	60120	153642	436190

$\operatorname{tr} a_n$ 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)[3]   # 85.6.a.d

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

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