Atlas › Modular forms › Weight 8 › Level 5 › 5.8.a › 5.8.a.b

Newform orbit 5.8.a.b

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 20q^{2} + 20q^{3} + 96q^{4} - 250q^{5} - 1016q^{6} - 100q^{7} + 1440q^{8} + 5554q^{9} - 2500q^{10} + 4544q^{11} - 23360q^{12} + 3540q^{13} + 7512q^{14} - 2500q^{15} + 20352q^{16} - 27340q^{17} + 31220q^{18} + 38760q^{19} - 12000q^{20} - 69096q^{21} + 106240q^{22} - 124140q^{23} - 131520q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{19})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} - 40x - 3324$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$

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
$\operatorname{tr} a_p$	20	20	−250	−100	4544	3540	−27340	38760	−124140	−72260	306824	−123020	264364	423300	−105460	−2391580	−1120120	2257044	4516460	621784	4569060	4333040	−9793020	6025620	4609540

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.229 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]: [1, 5] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 3 = 1 x 3 (phi(ord eps) x dim S_k^new); 1 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(5, 8)[1]   # 5.8.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 5:8 1000