Atlas › Modular forms › Weight 4 › Level 87 › 87.4.a › 87.4.a.a

Newform orbit 87.4.a.a

Trace form

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

$2q - 5q^{2} + 6q^{3} + 5q^{4} - 11q^{5} - 15q^{6} - 24q^{7} - 15q^{8} + 18q^{9} + 53q^{10} - 49q^{11} + 15q^{12} - 11q^{13} - 25q^{14} - 33q^{15} + 57q^{16} - 220q^{17} - 45q^{18} - 63q^{19} - 155q^{20} - 72q^{21} + 80q^{22} - 82q^{23} - 45q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} + 16x - 4$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
29	$+1$

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
$\operatorname{tr} a_p$	−5	6	−11	−24	−49	−11	−220	−63	−82	−58	210	−89	707	331	−568	−334	−119	−280	−123	18	−508	−410	−206	−101	−2060

$\operatorname{tr} a_n$ 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, 3, 29, 87]: [1, 2, 9, 32] (certified at ell = 2147483549)
T = [(2, 1), (5, 1)]; h of degree 14 = 1 x 14 (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(87, 4)[0]   # 87.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 87:4 1000