Atlas › Modular forms › Weight 2 › Level 340 › 340.2.a › 340.2.a.b

Newform orbit 340.2.a.b

Trace form

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

$3q + 3q^{5} + 7q^{9} + 2q^{11} + 2q^{13} + 3q^{17} + 16q^{21} + 8q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{7}$ , whose minimal polynomial is

$x^{3} - 128x + 256$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 340$ )
5	$-1$
17	$-1$

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
$\operatorname{tr} a_p$	0	0	3	0	2	2	3	0	8	−2	10	−6	18	−10	−2	14	−16	−6	−26	14	−10	−14	−18	26	−2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.207 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, 4, 5, 10, 17, 20, 34, 68, 85, 170, 340]: [0, 1, 2, 1, 3, 2, 6, 6, 12, 10, 30, 60] (certified at ell = 2147483489)
T = [(3, 1), (7, 3)]; h of degree 4 = 1 x 4 (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(340, 2)[1]   # 340.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 340:2 1000