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

Newform orbit 138.4.a.f

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 6q^{2} + 9q^{3} + 12q^{4} + 20q^{5} + 18q^{6} + 10q^{7} + 24q^{8} + 27q^{9} + 40q^{10} + 20q^{11} + 36q^{12} - 2q^{13} + 20q^{14} + 60q^{15} + 48q^{16} + 32q^{17} + 54q^{18} - 26q^{19} + 80q^{20} + 30q^{21} + 40q^{22} - 69q^{23} + 72q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7}$ , whose minimal polynomial is

$x^{3} - 50x^{2} - 2084x + 61992$

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
$\operatorname{tr} a_p$	6	9	20	10	20	−2	32	−26	−69	−50	12	−90	−466	−274	−680	−408	−1244	−130	450	−1512	710	478	−1092	−372	2106

$\operatorname{tr} a_n$ 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)[5]   # 138.4.a.f

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

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