Embedded newform 43.10.g.a.9.3

• Label: 43.10.g.a.9.3
• 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			9.3
Character	\(\chi\)	\(=\)	43.9
Dual form			43.10.g.a.24.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-24.7402 - 31.0232i) q^{2} +(98.3493 + 14.8238i) q^{3} +(-236.433 + 1035.88i) q^{4} +(1493.82 + 1018.47i) q^{5} +(-1973.30 - 3417.86i) q^{6} +(2565.02 - 4442.74i) q^{7} +(19681.4 - 9478.07i) q^{8} +(-9355.69 - 2885.85i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + 56 q^{2} - 321 q^{3} - 16052 q^{4} - 695 q^{5} + 1274 q^{6} + 32650 q^{7} - 50022 q^{8} + 373611 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−24.7402	−	31.0232i	−1.09337	−	1.37105i	−0.922610	−	0.385734i	\(-0.873948\pi\)
							−0.170763	−	0.985312i	\(-0.554623\pi\)
\(3\)	98.3493	+	14.8238i	0.701013	+	0.105661i	0.489871	−	0.871795i	\(-0.337044\pi\)
							0.211141	+	0.977456i	\(0.432282\pi\)
\(5\)	1493.82	+	1018.47i	1.06889	+	0.728759i	0.964048	−	0.265729i	\(-0.0856127\pi\)
							0.104846	+	0.994489i	\(0.466565\pi\)
\(7\)	2565.02	−	4442.74i	0.403784	−	0.699374i	−0.590395	−	0.807114i	\(-0.701028\pi\)
							0.994179	+	0.107740i	\(0.0343614\pi\)
\(11\)	14278.8	+	62559.5i	0.294052	+	1.28833i	0.878830	+	0.477134i	\(0.158325\pi\)
							−0.584778	+	0.811193i	\(0.698818\pi\)
\(13\)	−3960.40	+	52847.9i	−0.0384587	+	0.513195i	0.944396	+	0.328811i	\(0.106648\pi\)
							−0.982854	+	0.184384i	\(0.940971\pi\)
\(17\)	−480196.	+	327392.i	−1.39444	+	0.950710i	−0.394948	+	0.918704i	\(0.629237\pi\)
							−0.999488	+	0.0320063i	\(0.989810\pi\)
\(19\)	−291988.	+	90066.5i	−0.514013	+	0.158552i	−0.540901	−	0.841086i	\(-0.681917\pi\)
							0.0268875	+	0.999638i	\(0.491440\pi\)
\(23\)	25984.6	−	24110.2i	0.0193616	−	0.0179649i	−0.670433	−	0.741970i	\(-0.733891\pi\)
							0.689794	+	0.724005i	\(0.257701\pi\)
\(29\)	−11109.6	+	1674.50i	−0.00291681	+	0.000439638i	−0.150501	−	0.988610i	\(-0.548089\pi\)
							0.147584	+	0.989050i	\(0.452850\pi\)
\(31\)	−2.80639e6	+	7.15055e6i	−0.545783	+	1.39063i	0.346029	+	0.938224i	\(0.387530\pi\)
							−0.891811	+	0.452408i	\(0.850565\pi\)
\(37\)	2.86709e6	+	4.96595e6i	0.251498	+	0.435607i	0.963938	−	0.266126i	\(-0.0857436\pi\)
							−0.712441	+	0.701732i	\(0.752410\pi\)
\(41\)	1.77715e7	+	2.22848e7i	0.982195	+	1.23163i	0.972793	+	0.231678i	\(0.0744215\pi\)
							0.00940256	+	0.999956i	\(0.497007\pi\)
\(43\)	−1.35031e7	−	1.78958e7i	−0.602316	−	0.798258i
							\(47\)	7.73861e6	−	3.39050e7i	0.231325	−	1.01350i	−0.717217	−	0.696850i	\(-0.754584\pi\)
0.948542	−	0.316651i	\(-0.102558\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.g.a.9.3	✓	384
43.24	even	21	inner	43.10.g.a.24.3	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.9.3	✓	384	1.1	even	1	trivial
43.10.g.a.24.3	yes	384	43.24	even	21	inner