Atlas › Modular forms › Weight 2 › Level 137 › 137.2.a › 137.2.a.b

Newform orbit 137.2.a.b

Trace form

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

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

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

$x^{7} - 3x^{6} - 17x^{5} + 58x^{4} + 22x^{3} - 112x^{2} - 54x + 7$

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
$\operatorname{tr} a_p$	0	3	−2	15	−3	12	−6	10	−3	−9	13	−2	−1	7	15	−8	−6	1	24	16	−1	15	21	−8	1

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.159 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, 137]: [0, 12] (certified at ell = 2147483249)
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(137, 2)[1]   # 137.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 137:2 1000