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

Newform orbit 87.4.a.c

Trace form

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

$5q + 3q^{2} - 15q^{3} + 29q^{4} - q^{5} - 9q^{6} + 4q^{7} + 96q^{8} + 45q^{9} + 169q^{10} - 5q^{11} - 87q^{12} + 97q^{13} - 12q^{14} + 3q^{15} + 265q^{16} + 86q^{17} + 27q^{18} + 115q^{19} + 145q^{20} - 12q^{21} + 273q^{22} - 98q^{23} - 288q^{24} + O(q^{25})$

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

$x^{5} - 2x^{4} - 636x^{3} + 3421x^{2} + 102980x - 835200$

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	−1	4	−5	97	86	115	−98	−145	−154	279	223	1181	78	944	−551	694	73	582	1846	786	−206	−69	1674

$\operatorname{tr} a_n$ 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)[2]   # 87.4.a.c

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

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