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

Newform orbit 85.4.a.g

Trace form

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

$5q + 2q^{2} - q^{3} + 34q^{4} + 25q^{5} - 5q^{6} + 10q^{7} + 30q^{8} - 30q^{9} + 10q^{10} + 126q^{11} + 15q^{12} + 83q^{13} + 90q^{14} - 5q^{15} + 322q^{16} + 85q^{17} - 97q^{18} + 55q^{19} + 170q^{20} + 6q^{21} - 240q^{22} - 2q^{23} + 155q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} + x^{4} - 494x^{3} + 1804x^{2} + 34324x - 106120$

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
$\operatorname{tr} a_p$	2	−1	25	10	126	83	85	55	−2	195	97	−476	298	168	1095	−1035	1163	−351	1064	1393	−163	750	270	1179	−2745

$\operatorname{tr} a_n$ 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)[6]   # 85.4.a.g

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

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