Atlas › Modular forms › Weight 4 › Level 146 › 146.4.a › 146.4.a.e

Newform orbit 146.4.a.e

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q + 14q^{2} + 9q^{3} + 28q^{4} + 21q^{5} + 18q^{6} + 28q^{7} + 56q^{8} + 110q^{9} + 42q^{10} + 105q^{11} + 36q^{12} + 95q^{13} + 56q^{14} + 86q^{15} + 112q^{16} + 218q^{17} + 220q^{18} - 4q^{19} + 84q^{20} - 103q^{21} + 210q^{22} - 85q^{23} + 72q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{7} - 72x^{6} - 1753x^{5} + 175336x^{4} + 218150x^{3} - 84020762x^{2} - 496498304x + 1285640968$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
73	$-1$

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
$\operatorname{tr} a_p$	14	9	21	28	105	95	218	−4	−85	140	−40	34	−490	−262	−670	−346	−850	693	222	−297	511	413	−1669	686	1569

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.123 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, 2, 73, 146]: [1, 2, 20, 58] (certified at ell = 2147482873)
T = [(3, 1), (5, 3)]; h of degree 18 = 1 x 18 (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(146, 4)[4]   # 146.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 146:4 1000