Atlas › Modular forms › Weight 6 › Level 65 › 65.6.a › 65.6.a.d

Newform orbit 65.6.a.d

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 38q^{3} + 134q^{4} - 150q^{5} + 318q^{6} + 220q^{7} + 24q^{8} + 518q^{9} - 170q^{11} + 2238q^{12} + 1014q^{13} - 1440q^{14} - 950q^{15} + 3506q^{16} + 728q^{17} + 7788q^{18} + 1218q^{19} - 3350q^{20} - 396q^{21} + 5154q^{22} + 8954q^{23} + 12618q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} - 38x^{5} - 747x^{4} + 24900x^{3} + 251532x^{2} - 4153328x - 39408252$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
13	$-1$

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
$\operatorname{tr} a_p$	0	38	−150	220	−170	1014	728	1218	8954	8364	2862	13840	2248	3822	−10860	6584	−28874	972	60040	−74174	48960	214508	123260	94028	246284

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.243 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, 5, 13, 65]: [1, 3, 7, 36] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 20 = 1 x 20 (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(65, 6)[3]   # 65.6.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 65:6 1000