Atlas › Modular forms › Weight 4 › Level 78 › 78.4.a › 78.4.a.a

Newform orbit 78.4.a.a

$q$ -expansion

$q - 2q^{2} - 3q^{3} + 4q^{4} - 16q^{5} + 6q^{6} + 28q^{7} - 8q^{8} + 9q^{9} + 32q^{10} + 34q^{11} - 12q^{12} - 13q^{13} - 56q^{14} + 48q^{15} + 16q^{16} + 138q^{17} - 18q^{18} + 108q^{19} - 64q^{20} - 84q^{21} - 68q^{22} - 52q^{23} + 24q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{17}$ is a root of

$x - 1204$

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 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	−3	−16	28	34	−13	138	108	−52	−190	−176	342	240	−140	454	198	−154	34	−656	550	614	8	762	−444	1022

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ 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 0.133 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, 2, 5, 5, 13, 16, 46] (certified at ell = 2147483629)
T = [(5, 1), (7, 3), (11, 5), (17, 7)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 3 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, 4)[0]   # 78.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 78:4 1000