Atlas › Modular forms › Weight 6 › Level 87 › 87.6.a › 87.6.a.c

Newform orbit 87.6.a.c

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q + 13q^{2} + 63q^{3} + 145q^{4} + 64q^{5} + 117q^{6} + 168q^{7} + 588q^{8} + 567q^{9} + 799q^{10} + 832q^{11} + 1305q^{12} - 222q^{13} + 1526q^{14} + 576q^{15} + 1457q^{16} + 1782q^{17} + 1053q^{18} + 174q^{19} + 945q^{20} + 1512q^{21} - 5419q^{22} + 9884q^{23} + 5292q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{7} - 77x^{6} - 14950x^{5} + 1178357x^{4} + 36418217x^{3} - 2773679148x^{2} - 20552192336x + 1221511516352$

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
$\operatorname{tr} a_p$	13	63	64	168	832	−222	1782	174	9884	−5887	−384	18042	3826	−7366	27630	−35098	4504	−100326	−110236	69680	−123402	−19120	−7160	278306	92162

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.263 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, 3, 13, 52] (certified at ell = 2147483549)
T = [(2, 1), (5, 1)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 6 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, 6)[2]   # 87.6.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 87:6 1000