Atlas › Modular forms › Weight 4 › Level 138 › 138.4.a › 138.4.a.b

Newform orbit 138.4.a.b

$q$ -expansion

$q - 2q^{2} + 3q^{3} + 4q^{4} + 2q^{5} - 6q^{6} - 32q^{7} - 8q^{8} + 9q^{9} - 4q^{10} - 48q^{11} + 12q^{12} + 22q^{13} + 64q^{14} + 6q^{15} + 16q^{16} + 42q^{17} - 18q^{18} - 144q^{19} + 8q^{20} - 96q^{21} + 96q^{22} - 23q^{23} - 24q^{24} + O(q^{25})$

Coefficient field

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

$x + 94$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	$-1$
23	$+1$

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
$\operatorname{tr} a_p$	−2	3	2	−32	−48	22	42	−144	−23	174	−304	−318	74	192	392	−734	156	706	192	624	−406	696	−800	−102	−918

$\operatorname{tr} a_n$ 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 138, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.14 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, 23, 46, 69, 138]: [1, 2, 2, 5, 7, 20, 26, 76] (certified at ell = 2147483647)
T = [(5, 1), (7, 3)]; h of degree 10 = 1 x 10 (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(138, 4)[1]   # 138.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 138:4 1000