Atlas › Modular forms › Weight 6 › Level 46 › 46.6.a › 46.6.a.a

Newform orbit 46.6.a.a

Trace form

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

$2q - 8q^{2} + 6q^{3} + 32q^{4} - 68q^{5} - 24q^{6} + 96q^{7} - 128q^{8} - 28q^{9} + 272q^{10} - 684q^{11} + 96q^{12} - 498q^{13} - 384q^{14} - 2404q^{15} + 512q^{16} - 2904q^{17} + 112q^{18} - 2428q^{19} - 1088q^{20} - 152q^{21} + 2736q^{22} - 1058q^{23} - 384q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{55})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 62x - 2559$

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 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
$\operatorname{tr} a_p$	−8	6	−68	96	−684	−498	−2904	−2428	−1058	2226	1222	−5356	886	26720	−20594	7412	3600	27132	44300	−22962	−3878	−42868	−36856	−23352	79232

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.234 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, 11, 32] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 3 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, 6)[0]   # 46.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 46:6 1000