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

Newform orbit 87.4.a.d

Trace form

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

$5q + 3q^{2} + 15q^{3} + 29q^{4} + 29q^{5} + 9q^{6} + 4q^{7} + 18q^{8} + 45q^{9} - 97q^{10} + 69q^{11} + 87q^{12} + 29q^{13} + 126q^{14} + 87q^{15} + 41q^{16} + 72q^{17} + 27q^{18} - 47q^{19} + 159q^{20} + 12q^{21} - 47q^{22} + 250q^{23} + 54q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} - 32x^{4} + 196x^{3} + 1807x^{2} - 10718x - 46936$

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 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	15	29	4	69	29	72	−47	250	145	−318	−27	99	−333	340	176	1507	−402	453	818	−462	698	1246	853	−382

$\operatorname{tr} a_n$ 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)[3]   # 87.4.a.d

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

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