Atlas › Modular forms › Weight 2 › Level 127 › 127.2.a › 127.2.a.b

Newform orbit 127.2.a.b

Trace form

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

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

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

$x^{7} - 5x^{6} - 8x^{5} + 67x^{4} - 83x^{3} - 2x^{2} + 28x - 7$

Atkin–Lehner signs

$p$	$w_p$
127	$-1$

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
$\operatorname{tr} a_p$	2	3	8	−3	0	−1	24	−5	−1	−7	−8	−6	14	−1	25	29	−12	7	−25	7	13	−23	26	13	−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 0.16 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, 127]: [0, 11] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 1 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(127, 2)[1]   # 127.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 127:2 1000