Atlas › Modular forms › Weight 2 › Level 652 › 652.2.a › 652.2.a.b

Newform orbit 652.2.a.b

Trace form

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

$5q + 5q^{3} - q^{5} + 8q^{7} + 12q^{9} + 6q^{11} - 2q^{15} + 5q^{17} - 5q^{19} - q^{21} + 10q^{23} + O(q^{25})$

Coefficient field

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

$x^{5} - 4x^{4} - 15x^{3} + 54x^{2} + 36x - 79$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 652$ )
163	$+1$

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	5	−1	8	6	0	5	−5	10	19	5	−11	23	8	15	−12	7	−10	−2	−1	−20	−24	34	−4	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 0.212 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, 163, 326, 652]: [0, 1, 2, 14, 43, 85] (certified at ell = 2147483179)
T = [(3, 1), (5, 1)]; h of degree 13 = 1 x 13 (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(652, 2)[1]   # 652.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 652:2 1000