Atlas › Modular forms › Weight 4 › Level 139 › 139.4.a › 139.4.a.a

Newform orbit 139.4.a.a

Trace form

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

$2q + 2q^{2} - 4q^{3} + 2q^{4} + 10q^{5} - 28q^{6} - 34q^{7} + 18q^{8} - 10q^{9} - 14q^{10} + 42q^{11} - 52q^{12} - 150q^{13} - 26q^{14} + 16q^{15} - 78q^{16} + 100q^{17} + 86q^{18} - 220q^{19} - 38q^{20} + 56q^{21} - 38q^{22} + 80q^{23} + 84q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{2})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} + 2x - 1$

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 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	−4	10	−34	42	−150	100	−220	80	−170	−66	−216	624	120	480	−268	92	44	470	−6	−228	−1122	666	−1286	−592

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.112 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, 139]: [1, 36] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 34 = 1 x 34 (phi(ord eps) x dim S_k^new); 5 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, 4)[0]   # 139.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 139:4 1000