Atlas › Modular forms › Weight 6 › Level 53 › 53.6.a › 53.6.a.b

Newform orbit 53.6.a.b

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q + 8q^{2} + 43q^{3} + 224q^{4} + 86q^{5} + 129q^{6} + 380q^{7} + 294q^{8} + 1021q^{9} + 717q^{10} + 444q^{11} + 555q^{12} + 2027q^{13} + 4572q^{14} + 2992q^{15} + 5036q^{16} + 2201q^{17} + 8943q^{18} + 7831q^{19} + 3859q^{20} + 3846q^{21} + 2939q^{22} - 493q^{23} - 10927q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{12} - 51x^{11} - 1101x^{10} + 77969x^{9} + 317613x^{8} - 43924901x^{7} + 27309005x^{6} + 11000703475x^{5} - 26586387006x^{4} - 1176072939536x^{3} + 4474638313036x^{2} + 43254516477864x - 211116842821728$

Atkin–Lehner signs

$p$	$w_p$
53	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

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
$\operatorname{tr} a_p$	8	43	86	380	444	2027	2201	7831	−493	−14449	6470	1127	−9320	33394	−8290	33708	49794	163400	105582	41051	148384	137581	148855	−39402	−173291

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.228 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, 53]: [1, 23] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 4 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(53, 6)[1]   # 53.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 53:6 1000