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

Newform orbit 65.6.a.b

Trace form

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

$3q - 2q^{2} - 16q^{3} - 10q^{4} + 75q^{5} + 22q^{6} - 208q^{7} + 144q^{8} + 143q^{9} - 50q^{10} - 616q^{11} - 722q^{12} + 507q^{13} - 1284q^{14} - 400q^{15} - 1870q^{16} - 794q^{17} - 2150q^{18} - 3360q^{19} - 250q^{20} - 3792q^{21} + 2386q^{22} - 7076q^{23} + 522q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 18x^{2} - 339x - 2016$

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 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
$\operatorname{tr} a_p$	−2	−16	75	−208	−616	507	−794	−3360	−7076	2306	−6380	16010	8286	6816	7640	9618	4752	−86894	−91620	−123252	−78634	−110696	−18908	−112210	168806

$\operatorname{tr} a_n$ 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)[1]   # 65.6.a.b

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

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