Atlas › Modular forms › Weight 8 › Level 51 › 51.8.a › 51.8.a.d

Newform orbit 51.8.a.d

Trace form

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

$4q + 2q^{2} + 108q^{3} + 104q^{4} + 257q^{5} + 54q^{6} + 2244q^{7} + 384q^{8} + 2916q^{9} - 6632q^{10} + 9529q^{11} + 2808q^{12} + 21779q^{13} - 4184q^{14} + 6939q^{15} + 5592q^{16} + 19652q^{17} + 1458q^{18} - 72755q^{19} + 90108q^{20} + 60588q^{21} + 81052q^{22} + 83285q^{23} + 10368q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{4} - 259x^{3} - 263303x^{2} + 69197553x + 2890266480$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
17	$-1$

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
$\operatorname{tr} a_p$	2	108	257	2244	9529	21779	19652	−72755	83285	224304	11182	−538790	−859485	−589941	778886	72820	405066	605342	−2006200	5956416	−5116236	−8541226	11446778	17612798	22609850

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.294 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, 3, 17, 51]: [1, 3, 12, 44] (certified at ell = 2147483489)
T = [(2, 1), (5, 1)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 6 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(51, 8)[3]   # 51.8.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 51:8 1000