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

Newform orbit 55.8.a.b

Trace form

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

$5q - 3q^{2} + 105q^{4} + 625q^{5} - 1187q^{6} - 890q^{7} - 4839q^{8} + 2731q^{9} - 375q^{10} - 6655q^{11} + 16895q^{12} - 18494q^{13} + 10657q^{14} + 22913q^{16} - 38018q^{17} - 73364q^{18} - 61042q^{19} + 13125q^{20} - 195408q^{21} + 3993q^{22} - 144268q^{23} - 255221q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{5} + 3x^{4} - 6014x^{3} + 122628x^{2} + 3963224x - 86900768$

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 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
$\operatorname{tr} a_p$	−3	0	625	−890	−6655	−18494	−38018	−61042	−144268	−74518	−55496	−68764	−1265666	−435368	555576	−1440556	9145744	2733970	−3578412	−2377688	−8183798	4521624	−2281352	−16239504	−16909630

$\operatorname{tr} a_n$ 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)[1]   # 55.8.a.b

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

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