Atlas › Modular forms › Weight 2 › Level 344 › 344.2.a › 344.2.a.d

Newform orbit 344.2.a.d

Trace form

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

$5q - q^{3} + q^{5} + 2q^{7} + 12q^{9} + 2q^{11} + 8q^{13} + 15q^{17} + 7q^{19} + 10q^{21} - 3q^{23} + O(q^{25})$

Coefficient field

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

$x^{5} - 2x^{4} - 205x^{3} + 814x^{2} + 8556x - 40648$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 344$ )
43	$+1$

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	−1	1	2	2	8	15	7	−3	1	7	−7	11	−5	−17	−10	−16	0	14	−16	−4	−1	−8	18	25

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.197 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, 8, 43, 86, 172, 344]: [0, 1, 2, 3, 4, 13, 25, 48] (certified at ell = 2147483647)
T = [(3, 1), (5, 3)]; h of degree 11 = 1 x 11 (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(344, 2)[3]   # 344.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 344:2 1000