Atlas › Modular forms › Weight 2 › Level 388 › 388.2.a › 388.2.a.a

Newform orbit 388.2.a.a

Trace form

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

$3q - 2q^{3} - 5q^{5} + q^{7} - 3q^{9} - 7q^{11} - 4q^{13} + q^{15} - 11q^{17} - 3q^{19} - 10q^{21} - 2q^{23} + O(q^{25})$

Coefficient field

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

$x^{3} + 17x^{2} + 80x + 71$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 388$ )
97	$-1$

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	−2	−5	1	−7	−4	−11	−3	−2	−15	6	−4	−7	3	11	−10	1	3	−1	13	−5	12	−14	8	3

$\operatorname{tr} a_n$ 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, 2, 4, 97, 194, 388]: [0, 1, 2, 8, 26, 52] (certified at ell = 2147483137)
T = [(3, 1), (5, 3)]; h of degree 8 = 1 x 8 (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(388, 2)[0]   # 388.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 388:2 1000