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

Newform orbit 55.8.a.c

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - 5q^{2} + 251q^{4} - 750q^{5} - 547q^{6} + 550q^{7} + 705q^{8} + 4584q^{9} + 625q^{10} - 7986q^{11} + 33905q^{12} - 6960q^{13} - 2243q^{14} + 3587q^{16} + 45720q^{17} + 2060q^{18} - 8518q^{19} - 31375q^{20} + 218592q^{21} + 6655q^{22} + 53960q^{23} + 286659q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} + 5x^{5} - 8803x^{4} - 119035x^{3} + 17449820x^{2} + 495618500x + 2429642064$

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
$\operatorname{tr} a_p$	−5	0	−750	550	−7986	−6960	45720	−8518	53960	161976	−140144	130190	1064496	−654360	783060	1188190	941180	−2983480	−5828900	−4046000	4345620	2710016	22519040	16372146	−10338420

$\operatorname{tr} a_n$ 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)[2]   # 55.8.a.c

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

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