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

Newform orbit 65.4.a.e

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 8q^{3} + 34q^{4} + 25q^{5} - 42q^{6} + 38q^{7} - 12q^{8} + 65q^{9} + 46q^{11} + 138q^{12} + 65q^{13} + 60q^{14} + 40q^{15} + 322q^{16} - 178q^{17} - 324q^{18} + 118q^{19} + 170q^{20} + 48q^{21} - 438q^{22} - 136q^{23} - 1098q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{5} - 24x^{4} - 523x^{3} + 14542x^{2} + 8712x - 1165956$

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	8	25	38	46	65	−178	118	−136	−102	406	−34	322	40	270	530	778	118	−118	−518	1102	2148	−1102	−1678	−498

$\operatorname{tr} a_n$ 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)[4]   # 65.4.a.e

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

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