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

Newform orbit 146.4.a.d

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - 12q^{2} + q^{3} + 24q^{4} - 9q^{5} - 2q^{6} + 28q^{7} - 48q^{8} + 33q^{9} + 18q^{10} + 17q^{11} + 4q^{12} + 65q^{13} - 56q^{14} + 86q^{15} + 96q^{16} - 48q^{17} - 66q^{18} + 212q^{19} - 36q^{20} + 419q^{21} - 34q^{22} + 171q^{23} - 8q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{6} + 26x^{5} - 7033x^{4} - 130020x^{3} + 12472798x^{2} + 60849762x - 4211157314$

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
$\operatorname{tr} a_p$	−12	1	−9	28	17	65	−48	212	171	472	596	412	−620	616	−266	−1614	−1852	327	90	−165	−438	1845	−93	−1156	−433

$\operatorname{tr} a_n$ 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)[3]   # 146.4.a.d

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

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