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

Newform orbit 55.8.a.a

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 21q^{2} + 189q^{4} - 500q^{5} + 411q^{6} + 1262q^{7} - 2367q^{8} - 2706q^{9} + 2625q^{10} + 5324q^{11} + 4353q^{12} + 2696q^{13} - 16259q^{14} + 7633q^{16} - 24824q^{17} - 13470q^{18} - 22730q^{19} - 23625q^{20} + 5544q^{21} - 27951q^{22} - 160q^{23} - 179787q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} + 21x^{3} - 3562x^{2} + 67756x - 191848$

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
$\operatorname{tr} a_p$	−21	0	−500	1262	5324	2696	−24824	−22730	−160	−433924	−517576	523762	−277004	−251596	−711180	−940654	1014028	3634268	−163644	−5547896	−3614404	−8995368	−5010356	−8759946	11731552

$\operatorname{tr} a_n$ 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)[0]   # 55.8.a.a

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

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