Atlas › Modular forms › Weight 4 › Level 145 › 145.4.a › 145.4.a.e

Newform orbit 145.4.a.e

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + 5q^{2} + 17q^{3} + 41q^{4} + 40q^{5} + 3q^{6} + 33q^{7} + 21q^{8} + 91q^{9} + 25q^{10} + 96q^{11} + 203q^{12} + 39q^{13} - 109q^{14} + 85q^{15} + 45q^{16} + 93q^{17} - 24q^{18} + 8q^{19} + 205q^{20} + 64q^{21} + 42q^{22} + 295q^{23} - 93q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} - 22x^{7} + 33x^{6} + 1851x^{5} - 7462x^{4} - 50267x^{3} + 242434x^{2} + 442092x - 2296348$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
29	$-1$

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
$\operatorname{tr} a_p$	5	17	40	33	96	39	93	8	295	232	41	106	−690	539	806	−159	397	−709	−474	1264	1941	−253	876	1046	−221

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.116 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, 29, 145]: [1, 3, 9, 48] (certified at ell = 2147483549)
T = [(2, 1), (3, 1)]; h of degree 28 = 1 x 28 (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(145, 4)[4]   # 145.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 145:4 1000