Embedded newform 43.10.g.a.9.17

• Label: 43.10.g.a.9.17
• 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			9.17
Character	\(\chi\)	\(=\)	43.9
Dual form			43.10.g.a.24.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.815943 + 1.02316i) q^{2} +(-202.451 - 30.5146i) q^{3} +(113.550 - 497.493i) q^{4} +(626.829 + 427.365i) q^{5} +(-133.967 - 232.038i) q^{6} +(-4732.25 + 8196.49i) q^{7} +(1205.35 - 580.466i) q^{8} +(21246.7 + 6553.73i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + 56 q^{2} - 321 q^{3} - 16052 q^{4} - 695 q^{5} + 1274 q^{6} + 32650 q^{7} - 50022 q^{8} + 373611 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.815943	+	1.02316i	0.0360599	+	0.0452177i	0.799534	−	0.600620i	\(-0.205080\pi\)
							−0.763475	+	0.645838i	\(0.776508\pi\)
\(3\)	−202.451	−	30.5146i	−1.44303	−	0.217501i	−0.619596	−	0.784921i	\(-0.712704\pi\)
							−0.823429	+	0.567419i	\(0.807942\pi\)
\(5\)	626.829	+	427.365i	0.448522	+	0.305797i	0.766434	−	0.642323i	\(-0.222029\pi\)
							−0.317912	+	0.948120i	\(0.602982\pi\)
\(7\)	−4732.25	+	8196.49i	−0.744949	+	1.29029i	0.205270	+	0.978705i	\(0.434193\pi\)
							−0.950219	+	0.311584i	\(0.899141\pi\)
\(11\)	1333.44	+	5842.19i	0.0274604	+	0.120312i	0.986800	−	0.161941i	\(-0.0517754\pi\)
							−0.959340	+	0.282253i	\(0.908918\pi\)
\(13\)	−1711.08	+	22832.8i	−0.0166159	+	0.221724i	0.982714	+	0.185129i	\(0.0592703\pi\)
							−0.999330	+	0.0365953i	\(0.988349\pi\)
\(17\)	−168970.	+	115202.i	−0.490670	+	0.334533i	−0.783249	−	0.621708i	\(-0.786439\pi\)
							0.292579	+	0.956241i	\(0.405487\pi\)
\(19\)	747105.	−	230451.i	1.31520	−	0.405684i	0.443739	−	0.896156i	\(-0.353652\pi\)
							0.871457	+	0.490472i	\(0.163175\pi\)
\(23\)	729748.	−	677107.i	0.543748	−	0.504524i	−0.359789	−	0.933034i	\(-0.617151\pi\)
							0.903537	+	0.428509i	\(0.140961\pi\)
\(29\)	754579.	−	113734.i	0.198113	−	0.0298608i	−0.0492359	−	0.998787i	\(-0.515679\pi\)
							0.247349	+	0.968926i	\(0.420441\pi\)
\(31\)	1.84902e6	−	4.71122e6i	0.359595	−	0.916232i	−0.630546	−	0.776152i	\(-0.717169\pi\)
							0.990140	−	0.140080i	\(-0.0447359\pi\)
\(37\)	4.00091e6	+	6.92979e6i	0.350955	+	0.607872i	0.986417	−	0.164260i	\(-0.0525237\pi\)
							−0.635462	+	0.772132i	\(0.719190\pi\)
\(41\)	−1.38820e7	−	1.74074e7i	−0.767227	−	0.962072i	0.232718	−	0.972544i	\(-0.425238\pi\)
							−0.999945	+	0.0104722i	\(0.996667\pi\)
\(43\)	1.61899e7	−	1.55074e7i	0.722165	−	0.691721i
							\(47\)	7.15556e6	−	3.13506e7i	0.213896	−	0.937141i	−0.747994	−	0.663705i	\(-0.768983\pi\)
0.961890	−	0.273435i	\(-0.0881600\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.g.a.9.17	✓	384
43.24	even	21	inner	43.10.g.a.24.17	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.9.17	✓	384	1.1	even	1	trivial
43.10.g.a.24.17	yes	384	43.24	even	21	inner