Atlas › Modular forms › Weight 2 › Level 339 › 339.2.a › 339.2.a.h

Newform orbit 339.2.a.h

Trace form

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

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

Coefficient field

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

$x^{5} + 5x^{4} - 226x^{3} - 1086x^{2} + 12721x + 59401$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
113	$-1$

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
$\operatorname{tr} a_p$	1	−5	−2	−5	4	−3	−1	8	−11	22	7	12	−6	10	−8	10	7	22	−26	0	0	8	−38	36	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 0.192 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, 113, 339]: [0, 1, 10, 40] (certified at ell = 2147480161)
T = [(2, 1), (5, 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(339, 2)[7]   # 339.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 339:2 1000