Atlas › Modular forms › Weight 4 › Level 85 › 85.4.a › 85.4.a.f

Newform orbit 85.4.a.f

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 3q^{2} + 9q^{3} - q^{4} - 15q^{5} + 33q^{6} + 34q^{7} + 39q^{8} + 60q^{9} - 15q^{10} + 52q^{11} + 17q^{12} + 19q^{13} - 2q^{14} - 45q^{15} + 59q^{16} - 51q^{17} - 153q^{19} + 5q^{20} + 286q^{21} - 64q^{22} + 162q^{23} - 39q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} - 30x^{2} - 295x + 9832$

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
$\operatorname{tr} a_p$	3	9	−15	34	52	19	−51	−153	162	45	−67	−308	498	−246	101	893	355	1019	334	313	639	−92	2736	1623	−475

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.106 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, 6, 30] (certified at ell = 2147483489)
T = [(2, 1), (3, 3)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 3 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, 4)[5]   # 85.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 85:4 1000