Atlas › Modular forms › Weight 4 › Level 46 › 46.4.a › 46.4.a.d

Newform orbit 46.4.a.d

Trace form

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

$2q + 4q^{2} + 3q^{3} + 8q^{4} + 10q^{5} + 6q^{6} + 12q^{7} + 16q^{8} - 13q^{9} + 20q^{10} - 12q^{11} + 12q^{12} - 51q^{13} + 24q^{14} - 58q^{15} + 32q^{16} - 66q^{17} - 26q^{18} - 16q^{19} + 40q^{20} - 128q^{21} - 24q^{22} + 46q^{23} + 24q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{73})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{2} - 33x - 184$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
23	$-1$

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
$\operatorname{tr} a_p$	4	3	10	12	−12	−51	−66	−16	46	−57	−17	206	373	314	859	50	612	−1062	−844	399	−1277	122	1802	2046	−910

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.102 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, 23, 46]: [1, 2, 7, 20] (certified at ell = 2147483647)
T = [(3, 1), (5, 3)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 2 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(46, 4)[3]   # 46.4.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 46:4 1000