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

Newform orbit 85.6.a.b

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q + 9q^{2} + 18q^{3} + 139q^{4} - 175q^{5} - 41q^{6} + 50q^{7} + 705q^{8} + 825q^{9} - 225q^{10} + 902q^{11} + 2359q^{12} + 172q^{13} + 1422q^{14} - 450q^{15} + 2515q^{16} + 2023q^{17} + 1058q^{18} + 2060q^{19} - 3475q^{20} - 3400q^{21} + 9144q^{22} - 790q^{23} + 14733q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{7} - 27x^{6} - 1039x^{5} + 20883x^{4} + 349806x^{3} - 4638840x^{2} - 34543584x + 335086560$

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
$\operatorname{tr} a_p$	9	18	−175	50	902	172	2023	2060	−790	6836	2734	−5766	−8890	6396	53816	97468	28188	8864	9016	−24034	79708	−267954	159812	178300	295464

$\operatorname{tr} a_n$ 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)[1]   # 85.6.a.b

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

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