Atlas › Modular forms › Weight 2 › Level 159 › 159.2.a › 159.2.a.a

Newform orbit 159.2.a.a

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{4} - 5x^{3} + 4x^{2} + 8x - 9$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
53	$+1$

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
$\operatorname{tr} a_p$	3	4	2	−4	6	−6	10	−6	2	6	−12	−10	12	−16	0	−4	10	14	−6	−8	10	10	12	−4	6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.085 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, 3, 53, 159]: [0, 1, 5, 20] (certified at ell = 2147483053)
T = [(2, 1), (5, 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(159, 2)[0]   # 159.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 159:2 1000