Atlas › Modular forms › Weight 2 › Level 119 › 119.2.a › 119.2.a.b

Newform orbit 119.2.a.b

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{5} - 22x^{3} + 46x^{2} - 16x - 11$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
17	$-1$

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	−2	0	−5	−2	2	5	6	−10	−8	0	8	18	8	−10	4	8	22	16	−2	10	18	−12	20	12

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.083 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, 7, 17, 119]: [0, 1, 2, 14] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 9 = 1 x 9 (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(119, 2)[1]   # 119.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 119:2 1000