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

Newform orbit 53.6.a.a

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q - 4q^{2} - 47q^{3} + 80q^{4} - 14q^{5} - 87q^{6} - 404q^{7} - 282q^{8} + 292q^{9} - 683q^{10} - 524q^{11} - 2325q^{12} - 1353q^{13} - 4002q^{14} - 4000q^{15} - 4508q^{16} - 2847q^{17} - 4047q^{18} - 3893q^{19} - 2101q^{20} + 1208q^{21} - 33q^{22} - 4289q^{23} + 2841q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} + 51x^{8} - 36x^{7} - 28078x^{6} - 212939x^{5} + 4030367x^{4} + 42913558x^{3} - 92968836x^{2} - 1527177000x - 1276100320$

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 odd 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
$\operatorname{tr} a_p$	−4	−47	−14	−404	−524	−1353	−2847	−3893	−4289	1887	−5796	−317	6334	−52718	−22026	−25281	−78014	−74064	−142248	−81775	−140464	35299	−60065	190608	−187427

$\operatorname{tr} a_n$ 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)[0]   # 53.6.a.a

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

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