Atlas › Modular forms › Weight 6 › Level 52 › 52.6.a › 52.6.a.c

Newform orbit 52.6.a.c

Trace form

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

$3q + 12q^{3} + 8q^{5} + 138q^{7} + 1041q^{9} - 290q^{11} - 507q^{13} + 4812q^{15} + 2592q^{17} + 378q^{19} - 1158q^{21} + 3800q^{23} + O(q^{25})$

Coefficient field

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

$x^{3} - 20x^{2} - 14116x - 38128$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 52$ )
13	$+1$

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	12	8	138	−290	−507	2592	378	3800	5270	−16422	−2592	−26742	−7152	−110	52542	48374	77214	−23046	38110	27294	41388	−26526	−66282	−274674

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.236 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 engine's checks

dim M^+ at levels [1, 2, 4, 13, 26, 52]: [1, 2, 4, 7, 19, 38] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 5 = 1 x 5 (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(52, 6)[2]   # 52.6.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 52:6 1000