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

Newform orbit 85.4.a.d

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 4q^{2} - 2q^{3} - 2q^{4} + 10q^{5} + 10q^{6} + 2q^{7} + 12q^{8} - 46q^{9} - 20q^{10} - 38q^{11} - 22q^{12} - 80q^{13} - 58q^{14} - 10q^{15} - 14q^{16} - 34q^{17} + 80q^{18} - 180q^{19} - 10q^{20} - 56q^{21} + 154q^{22} - 42q^{23} - 18q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{3})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{2} + 10x - 23$

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
$\operatorname{tr} a_p$	−4	−2	10	2	−38	−80	−34	−180	−42	−216	70	352	344	−88	−348	−164	−1436	−76	−44	−1510	1088	−958	248	−204	1012

$\operatorname{tr} a_n$ 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)[3]   # 85.4.a.d

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

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