Atlas › Modular forms › Weight 2 › Level 139 › 139.2.a › 139.2.a.c

Newform orbit 139.2.a.c

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{7} + x^{6} - 21x^{5} + 2x^{4} + 97x^{3} + 4x^{2} - 140x - 72$

Atkin–Lehner signs

$p$	$w_p$
139	$-1$

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
$\operatorname{tr} a_p$	1	−2	11	−5	2	6	5	−10	−1	30	−20	6	19	−12	−3	38	−14	4	9	24	−5	8	−9	10	−5

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.163 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 139]: [0, 12] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 2 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(139, 2)[2]   # 139.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 139:2 1000