Embedded newform 43.10.g.a.9.13

• Label: 43.10.g.a.9.13
• Level: $43$
• Weight: $10$
• Character: 43.9
• Analytic conductor: $22.147$
• Analytic rank: $0$
• Dimension: $384$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(22.1465409550\)
Analytic rank:	\(0\)
Dimension:	\(384\)
Relative dimension:	\(32\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			9.13
Character	\(\chi\)	\(=\)	43.9
Dual form			43.10.g.a.24.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.76674 - 8.48522i) q^{2} +(43.2869 + 6.52445i) q^{3} +(87.7205 - 384.328i) q^{4} +(1337.26 + 911.727i) q^{5} +(-237.550 - 411.448i) q^{6} +(-3892.00 + 6741.15i) q^{7} +(-8861.15 + 4267.30i) q^{8} +(-16977.4 - 5236.82i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + 56 q^{2} - 321 q^{3} - 16052 q^{4} - 695 q^{5} + 1274 q^{6} + 32650 q^{7} - 50022 q^{8} + 373611 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{21}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.76674	−	8.48522i	−0.299051	−	0.374998i	0.609490	−	0.792793i	\(-0.291374\pi\)
							−0.908541	+	0.417796i	\(0.862803\pi\)
\(3\)	43.2869	+	6.52445i	0.308540	+	0.0465048i	0.301487	−	0.953470i	\(-0.402517\pi\)
							0.00705250	+	0.999975i	\(0.497755\pi\)
\(5\)	1337.26	+	911.727i	0.956863	+	0.652378i	0.937703	−	0.347438i	\(-0.112948\pi\)
							0.0191604	+	0.999816i	\(0.493901\pi\)
\(7\)	−3892.00	+	6741.15i	−0.612677	+	1.06119i	0.378110	+	0.925761i	\(0.376574\pi\)
							−0.990787	+	0.135428i	\(0.956759\pi\)
\(11\)	−2027.11	−	8881.33i	−0.0417455	−	0.182899i	0.949757	−	0.312989i	\(-0.101330\pi\)
							−0.991502	+	0.130090i	\(0.958473\pi\)
\(13\)	1495.77	−	19959.6i	0.0145251	−	0.193824i	−0.985221	−	0.171288i	\(-0.945207\pi\)
							0.999746	−	0.0225355i	\(-0.00717389\pi\)
\(17\)	82146.6	−	56006.6i	0.238545	−	0.162637i	−0.438148	−	0.898903i	\(-0.644365\pi\)
							0.676692	+	0.736266i	\(0.263413\pi\)
\(19\)	−994250.	+	306685.i	−1.75027	+	0.539886i	−0.993503	−	0.113803i	\(-0.963697\pi\)
							−0.756763	+	0.653689i	\(0.773220\pi\)
\(23\)	−78744.9	+	73064.5i	−0.0586742	+	0.0544417i	−0.708965	−	0.705244i	\(-0.750838\pi\)
							0.650291	+	0.759685i	\(0.274647\pi\)
\(29\)	−6.17764e6	+	931129.i	−1.62193	+	0.244466i	−0.896244	−	0.443562i	\(-0.853715\pi\)
							−0.725684	+	0.688028i	\(0.758477\pi\)
\(31\)	−460129.	+	1.17239e6i	−0.0894853	+	0.228005i	−0.968609	−	0.248588i	\(-0.920033\pi\)
							0.879124	+	0.476593i	\(0.158129\pi\)
\(37\)	8.11814e6	+	1.40610e7i	0.712113	+	1.23342i	0.964063	+	0.265676i	\(0.0855950\pi\)
							−0.251950	+	0.967740i	\(0.581072\pi\)
\(41\)	7.92102e6	+	9.93265e6i	0.437778	+	0.548956i	0.950956	−	0.309326i	\(-0.100103\pi\)
							−0.513178	+	0.858282i	\(0.671532\pi\)
\(43\)	−1.61203e7	+	1.55798e7i	−0.719059	+	0.694949i
							\(47\)	1.24215e7	−	5.44220e7i	0.371307	−	1.62680i	−0.351808	−	0.936072i	\(-0.614433\pi\)
0.723115	−	0.690728i	\(-0.242710\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.g.a.9.13	✓	384
43.24	even	21	inner	43.10.g.a.24.13	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.9.13	✓	384	1.1	even	1	trivial
43.10.g.a.24.13	yes	384	43.24	even	21	inner