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

Newform orbit 146.4.a.b

Trace form

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

$2q - 4q^{2} - 3q^{3} + 8q^{4} + 7q^{5} + 6q^{6} - 20q^{7} - 16q^{8} - 19q^{9} - 14q^{10} + 3q^{11} - 12q^{12} + 75q^{13} + 40q^{14} + 20q^{15} + 32q^{16} - 24q^{17} + 38q^{18} - 260q^{19} + 28q^{20} - 153q^{21} - 6q^{22} - 261q^{23} + 24q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{61})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{2} - 18x - 163$

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 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
$\operatorname{tr} a_p$	−4	−3	7	−20	3	75	−24	−260	−261	42	−62	−100	136	−744	136	706	−260	−741	562	−613	146	−711	1581	−408	−477

$\operatorname{tr} a_n$ 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)[1]   # 146.4.a.b

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

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