Atlas › Modular forms › Weight 4 › Level 65 › 65.4.a › 65.4.a.b

Newform orbit 65.4.a.b

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 2q^{2} - 4q^{3} - 2q^{4} + 10q^{5} - 8q^{6} - 20q^{7} - 6q^{8} - 34q^{9} - 10q^{10} - 104q^{11} + 28q^{12} - 26q^{13} - 4q^{14} - 20q^{15} - 62q^{16} + 52q^{17} + 82q^{18} - 10q^{20} + 64q^{21} + 116q^{22} - 12q^{23} + 96q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{6})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{2} + 14x + 25$

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	−4	10	−20	−104	−26	52	0	−12	16	−464	−88	−180	−364	308	748	−480	16	−116	−480	744	456	−500	268	−724

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.103 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, 5, 24] (certified at ell = 2147483629)
T = [(2, 1), (3, 3)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 3 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, 4)[1]   # 65.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 65:4 1000