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

Newform orbit 139.4.a.c

Trace form

The 20 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$20q + 7q^{2} + 7q^{3} + 101q^{4} + 63q^{5} + 14q^{6} + 8q^{7} + 84q^{8} + 293q^{9} + 57q^{10} + 127q^{11} + 21q^{12} + 83q^{13} + 253q^{14} + 208q^{15} + 449q^{16} + 130q^{17} + 105q^{18} + 97q^{19} + 471q^{20} + 400q^{21} + 297q^{22} + 424q^{23} + 300q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 20. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{20} - 14x^{19} - 463x^{18} + 6723x^{17} + 83878x^{16} - 1267130x^{15} - 8089567x^{14} + 126191976x^{13} + 465680358x^{12} - 7386765757x^{11} - 16712112798x^{10} + 263765245077x^{9} + 376971094319x^{8} - 5712463072437x^{7} - 5379648783268x^{6} + 71505565592670x^{5} + 52172618137780x^{4} - 463523850587436x^{3} - 378489393801248x^{2} + 1223336764923608x + 1345839519759872$

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
$\operatorname{tr} a_p$	7	7	63	8	127	83	130	97	424	1273	198	−222	433	−59	−275	1951	983	549	−1145	2006	−20	−266	595	2216	254

$\operatorname{tr} a_n$ 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)[2]   # 139.4.a.c

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

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