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

Newform orbit 87.4.a.b

Trace form

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

$2q - q^{2} - 6q^{3} + 5q^{4} - q^{5} + 3q^{6} - 24q^{7} - 15q^{8} + 18q^{9} - 61q^{10} + 25q^{11} - 15q^{12} - 71q^{13} - 29q^{14} + 3q^{15} - 135q^{16} - 98q^{17} - 9q^{18} - 21q^{19} + 59q^{20} + 72q^{21} + 8q^{22} - 62q^{23} + 45q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{41})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} + 2x - 40$

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
$\operatorname{tr} a_p$	−1	−6	−1	−24	25	−71	−98	−21	−62	58	66	−539	−101	−155	526	−698	455	44	−1551	126	−760	−158	−934	−691	532

$\operatorname{tr} a_n$ 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)[1]   # 87.4.a.b

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

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