Atlas › Modular forms › Weight 2 › Level 338 › 338.2.a › 338.2.a.h

Newform orbit 338.2.a.h

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{3} - 9x^{2} - 22x + 211$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
13	— ( $13^2 \mid 338$ )

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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	3	2	−4	3	0	5	1	0	−10	−16	−14	7	11	−2	0	−3	−4	21	−12	1	−18	−1	25	23

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.288 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, 2, 13, 26, 169, 338]: [0, 1, 1, 5, 15, 47] (certified at ell = 2147483053)
T = [(3, 1), (5, 3)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 2 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(338, 2)[7]   # 338.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 338:2 1000