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

Newform orbit 51.8.a.c

Trace form

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

$4q - 15q^{2} + 108q^{3} + 141q^{4} - 310q^{5} - 405q^{6} - 1036q^{7} - 3537q^{8} + 2916q^{9} - 2460q^{10} - 9922q^{11} + 3807q^{12} + 178q^{13} + 8974q^{14} - 8370q^{15} + 26065q^{16} - 19652q^{17} - 10935q^{18} - 17534q^{19} - 68140q^{20} - 27972q^{21} - 67712q^{22} - 92158q^{23} - 95499q^{24} + O(q^{25})$

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

$x^{4} + 325x^{3} - 48119x^{2} - 21143301x - 1301279706$

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 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
$\operatorname{tr} a_p$	−15	108	−310	−1036	−9922	178	−19652	−17534	−92158	−594312	−174816	−550236	−316650	474662	−1218044	855152	1088364	−274180	1614016	−5993068	9453112	−1195800	−7506204	−12703012	475652

$\operatorname{tr} a_n$ 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)[2]   # 51.8.a.c

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

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