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

Newform orbit 138.4.a.d

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 4q^{2} + 6q^{3} + 8q^{4} + 2q^{5} - 12q^{6} + 28q^{7} - 16q^{8} + 18q^{9} - 4q^{10} - 2q^{11} + 24q^{12} + 48q^{13} - 56q^{14} + 6q^{15} + 32q^{16} - 4q^{17} - 36q^{18} + 78q^{19} + 8q^{20} + 84q^{21} + 4q^{22} + 46q^{23} - 48q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{277})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7}$ , whose minimal polynomial is

$x^{2} - 86x + 1572$

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 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
$\operatorname{tr} a_p$	−4	6	2	28	−2	48	−4	78	46	184	308	370	12	186	−528	−926	−792	−42	−434	624	−352	−508	994	0	−784

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.14 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)[3]   # 138.4.a.d

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

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