Embedded newform 43.10.g.a.13.14

• Label: 43.10.g.a.13.14
• Level: $43$
• Weight: $10$
• Character: 43.13
• 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			13.14
Character	\(\chi\)	\(=\)	43.13
Dual form			43.10.g.a.10.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.01857 + 8.84394i) q^{2} +(52.4820 - 48.6961i) q^{3} +(387.155 + 186.444i) q^{4} +(1707.71 + 257.396i) q^{5} +(324.727 + 562.444i) q^{6} +(-3898.33 + 6752.11i) q^{7} +(-5326.23 + 6678.88i) q^{8} +(-1087.87 + 14516.6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + 56 q^{2} - 321 q^{3} - 16052 q^{4} - 695 q^{5} + 1274 q^{6} + 32650 q^{7} - 50022 q^{8} + 373611 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.01857	+	8.84394i	−0.0892092	+	0.390851i	−0.999745	−	0.0225792i	\(-0.992812\pi\)
							0.910536	+	0.413430i	\(0.135669\pi\)
\(3\)	52.4820	−	48.6961i	0.374080	−	0.347095i	−0.470620	−	0.882336i	\(-0.655970\pi\)
							0.844700	+	0.535241i	\(0.179779\pi\)
\(5\)	1707.71	+	257.396i	1.22194	+	0.184178i	0.728159	−	0.685408i	\(-0.240376\pi\)
							0.493782	+	0.869586i	\(0.335614\pi\)
\(7\)	−3898.33	+	6752.11i	−0.613674	+	1.06291i	0.376942	+	0.926237i	\(0.376976\pi\)
							−0.990616	+	0.136677i	\(0.956358\pi\)
\(11\)	22435.9	−	10804.6i	0.462037	−	0.222506i	−0.188359	−	0.982100i	\(-0.560317\pi\)
							0.650397	+	0.759595i	\(0.274603\pi\)
\(13\)	−24618.8	−	62727.8i	−0.239069	−	0.609137i	0.760104	−	0.649801i	\(-0.225148\pi\)
							−0.999173	+	0.0406642i	\(0.987053\pi\)
\(17\)	−142746.	+	21515.5i	−0.414519	+	0.0624787i	−0.352991	−	0.935627i	\(-0.614835\pi\)
							−0.0615278	+	0.998105i	\(0.519597\pi\)
\(19\)	12644.5	+	168729.i	0.0222593	+	0.297030i	0.997212	+	0.0746181i	\(0.0237738\pi\)
							−0.974953	+	0.222411i	\(0.928607\pi\)
\(23\)	−696310.	−	474736.i	−0.518833	−	0.353734i	0.275418	−	0.961325i	\(-0.411184\pi\)
							−0.794251	+	0.607590i	\(0.792136\pi\)
\(29\)	−76949.9	−	71399.0i	−0.0202030	−	0.0187457i	0.670007	−	0.742355i	\(-0.266291\pi\)
							−0.690210	+	0.723609i	\(0.742482\pi\)
\(31\)	−733438.	+	226236.i	−0.142638	+	0.0439981i	−0.365252	−	0.930909i	\(-0.619017\pi\)
							0.222614	+	0.974907i	\(0.428541\pi\)
\(37\)	1.09151e7	+	1.89055e7i	0.957457	+	1.65836i	0.728642	+	0.684895i	\(0.240152\pi\)
							0.228815	+	0.973470i	\(0.426515\pi\)
\(41\)	4.87441e6	−	2.13562e7i	0.269398	−	1.18031i	−0.641317	−	0.767276i	\(-0.721612\pi\)
							0.910715	−	0.413035i	\(-0.135531\pi\)
\(43\)	2.23812e7	+	1.29450e6i	0.998332	+	0.0577425i
							\(47\)	1.31680e7	+	6.34138e6i	0.393623	+	0.189559i	0.620216	−	0.784431i	\(-0.287045\pi\)
−0.226594	+	0.973989i	\(0.572759\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.g.a.13.14	yes	384
43.10	even	21	inner	43.10.g.a.10.14	✓	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.10.14	✓	384	43.10	even	21	inner
43.10.g.a.13.14	yes	384	1.1	even	1	trivial