Atlas › Modular forms › Weight 6 › Level 78 › 78.6.a › 78.6.a.f

Newform orbit 78.6.a.f

$q$ -expansion

$q + 4q^{2} + 9q^{3} + 16q^{4} - 88q^{5} + 36q^{6} - 176q^{7} + 64q^{8} + 81q^{9} - 352q^{10} - 674q^{11} + 144q^{12} - 169q^{13} - 704q^{14} - 792q^{15} + 256q^{16} + 1122q^{17} + 324q^{18} + 2112q^{19} - 1408q^{20} - 1584q^{21} - 2696q^{22} - 1900q^{23} + 576q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 3T_{7}$ is a root of

$x + 616$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	$-1$
13	$+1$

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
$\operatorname{tr} a_p$	4	9	−88	−176	−674	−169	1122	2112	−1900	−3574	−8324	−4002	−1080	6964	20338	−29130	−37726	22378	20164	−30038	31382	82208	−71298	−45156	−101410

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ for the primes $p < 1000$ not dividing 78, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.301 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, 2, 3, 6, 13, 26, 39, 78]: [1, 2, 3, 7, 7, 19, 26, 74] (certified at ell = 2147483629)
T = [(5, 1), (7, 3)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 4 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(78, 6)[5]   # 78.6.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 78:6 1000