Atlas › Modular forms › Weight 4 › Level 137 › 137.4.a › 137.4.a.b

Newform orbit 137.4.a.b

Trace form

The 19 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$19q + 6q^{2} + 18q^{3} + 92q^{4} + 10q^{5} + 5q^{6} + 130q^{7} + 54q^{8} + 205q^{9} + 109q^{10} + 16q^{11} + 176q^{12} + 178q^{13} + 40q^{14} + 260q^{15} + 468q^{16} + 126q^{17} + 279q^{18} + 474q^{19} + 55q^{20} + 190q^{21} + 239q^{22} + 126q^{23} + 68q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 19. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{19} - 24x^{18} - 198x^{17} + 8079x^{16} + 256x^{15} - 1107317x^{14} + 2827761x^{13} + 79643772x^{12} - 294683886x^{11} - 3224481899x^{10} + 13697286267x^{9} + 73042813204x^{8} - 328674738677x^{7} - 858677051430x^{6} + 4018694634960x^{5} + 4227866187587x^{4} - 22551343878518x^{3} - 3528554239274x^{2} + 42735267406066x - 10511080394762$

Atkin–Lehner signs

$p$	$w_p$
137	$+1$

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	18	10	130	16	178	126	474	126	286	1048	772	−680	622	350	−1024	374	296	1928	−568	3300	1946	136	258	3428

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.111 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, 137]: [1, 36] (certified at ell = 2147483249)
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(137, 4)[1]   # 137.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 137:4 1000