Embedded newform 43.10.g.a.24.13

• Label: 43.10.g.a.24.13
• Level: $43$
• Weight: $10$
• Character: 43.24
• 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			24.13
Character	\(\chi\)	\(=\)	43.24
Dual form			43.10.g.a.9.13

$q$-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 \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{20}{21}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.9.13	✓	384	43.9	even	21	inner
43.10.g.a.24.13	yes	384	1.1	even	1	trivial