Atlas › Modular forms › Weight 2 › Level 259 › 259.2.a › 259.2.a.f

Newform orbit 259.2.a.f

Trace form

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

$4q + 2q^{3} + 10q^{4} + 6q^{5} + 3q^{6} - 4q^{7} - 3q^{8} + 2q^{9} - 3q^{10} + 3q^{11} - 8q^{12} + 5q^{13} - 10q^{15} + 2q^{16} + 13q^{17} - 15q^{18} - 7q^{19} + 28q^{20} - 2q^{21} - 16q^{22} - 5q^{23} + 18q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{4} - 6x^{3} - 63x^{2} + 445x - 367$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
37	$-1$

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	2	6	−4	3	5	13	−7	−5	20	10	4	−11	−8	19	−1	−5	−3	−8	3	18	1	−8	39	−22

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.183 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, 37, 259]: [0, 1, 3, 26] (certified at ell = 2147483629)
T = [(2, 1), (3, 3)]; h of degree 19 = 1 x 19 (phi(ord eps) x dim S_k^new); 3 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(259, 2)[5]   # 259.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 259:2 1000