Atlas › Modular forms › Weight 8 › Level 55 › 55.8.a › 55.8.a.d

Newform orbit 55.8.a.d

Trace form

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

$7q + 29q^{2} + 108q^{3} + 555q^{4} + 875q^{5} + 635q^{6} + 1194q^{7} + 1305q^{8} + 10021q^{9} + 3625q^{10} + 9317q^{11} + 8079q^{12} + 8738q^{13} - 25699q^{14} + 13500q^{15} - 2685q^{16} + 30174q^{17} + 150118q^{18} + 49450q^{19} + 69375q^{20} - 20584q^{21} + 38599q^{22} + 197728q^{23} + 258981q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{7} - 137x^{6} - 4641x^{5} + 1037177x^{4} - 6526288x^{3} - 1828067104x^{2} + 16049594864x + 777518022512$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
11	$-1$

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
$\operatorname{tr} a_p$	29	108	875	1194	9317	8738	30174	49450	197728	262902	47976	1009384	−363270	306248	−152556	1291208	−1482584	587134	1355184	8812976	−5308222	−3776336	−6348152	10307028	40905374

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.292 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, 5, 11, 55]: [1, 5, 8, 44] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 22 = 1 x 22 (phi(ord eps) x dim S_k^new); 5 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(55, 8)[3]   # 55.8.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 55:8 1000