Embedded newform 43.10.g.a.13.16

• Label: 43.10.g.a.13.16
• Level: $43$
• Weight: $10$
• Character: 43.13
• Analytic conductor: $22.147$
• Analytic rank: $0$
• Dimension: $384$
• 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			13.16
Character	\(\chi\)	\(=\)	43.13
Dual form			43.10.g.a.10.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.699360 - 3.06409i) q^{2} +(39.6781 - 36.8159i) q^{3} +(452.396 + 217.863i) q^{4} +(-2404.07 - 362.355i) q^{5} +(-85.0582 - 147.325i) q^{6} +(715.663 - 1239.56i) q^{7} +(1987.24 - 2491.91i) q^{8} +(-1251.97 + 16706.4i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + 56 q^{2} - 321 q^{3} - 16052 q^{4} - 695 q^{5} + 1274 q^{6} + 32650 q^{7} - 50022 q^{8} + 373611 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{16}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.699360	−	3.06409i	0.0309076	−	0.135415i	−0.957120	−	0.289691i	\(-0.906447\pi\)
							0.988028	+	0.154276i	\(0.0493045\pi\)
\(3\)	39.6781	−	36.8159i	0.282817	−	0.262416i	−0.525989	−	0.850491i	\(-0.676305\pi\)
							0.808806	+	0.588075i	\(0.200114\pi\)
\(5\)	−2404.07	−	362.355i	−1.72021	−	0.259280i	−0.786574	−	0.617496i	\(-0.788147\pi\)
							−0.933639	+	0.358215i	\(0.883385\pi\)
\(7\)	715.663	−	1239.56i	0.112659	−	0.195132i	−0.804182	−	0.594383i	\(-0.797396\pi\)
							0.916842	+	0.399251i	\(0.130730\pi\)
\(11\)	57138.0	−	27516.2i	1.17668	−	0.566659i	0.259737	−	0.965679i	\(-0.416364\pi\)
							0.916942	+	0.399021i	\(0.130650\pi\)
\(13\)	27015.2	+	68833.5i	0.262339	+	0.668428i	0.999977	−	0.00676791i	\(-0.00215431\pi\)
							−0.737638	+	0.675196i	\(0.764059\pi\)
\(17\)	−49199.3	+	7415.61i	−0.142869	+	0.0215341i	−0.220088	−	0.975480i	\(-0.570634\pi\)
							0.0772182	+	0.997014i	\(0.475396\pi\)
\(19\)	−5644.10	−	75315.3i	−0.00993582	−	0.132584i	0.990030	−	0.140857i	\(-0.0449857\pi\)
							−0.999966	+	0.00827266i	\(0.997367\pi\)
\(23\)	1.03845e6	+	708000.i	0.773764	+	0.527543i	0.884631	−	0.466292i	\(-0.154410\pi\)
							−0.110867	+	0.993835i	\(0.535363\pi\)
\(29\)	2.78471e6	+	2.58383e6i	0.731121	+	0.678381i	0.955124	−	0.296208i	\(-0.0957221\pi\)
							−0.224003	+	0.974588i	\(0.571913\pi\)
\(31\)	3.35005e6	−	1.03336e6i	0.651515	−	0.200966i	0.0486585	−	0.998815i	\(-0.484505\pi\)
							0.602856	+	0.797850i	\(0.294029\pi\)
\(37\)	2.10772e6	+	3.65068e6i	0.184887	+	0.320233i	0.943538	−	0.331263i	\(-0.107475\pi\)
							−0.758652	+	0.651497i	\(0.774141\pi\)
\(41\)	1.89808e6	−	8.31605e6i	0.104903	−	0.459610i	−0.895005	−	0.446056i	\(-0.852828\pi\)
							0.999908	−	0.0135544i	\(-0.00431463\pi\)
\(43\)	6.52849e6	+	2.14469e7i	0.291209	+	0.956660i
							\(47\)	1.49572e6	+	720301.i	0.0447106	+	0.0215315i	0.456105	−	0.889926i	\(-0.349244\pi\)
−0.411395	+	0.911457i	\(0.634958\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.g.a.13.16	yes	384
43.10	even	21	inner	43.10.g.a.10.16	✓	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.10.16	✓	384	43.10	even	21	inner
43.10.g.a.13.16	yes	384	1.1	even	1	trivial