Embedded newform 43.10.g.a.10.14

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

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{21}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.10.14	✓	384	1.1	even	1	trivial
43.10.g.a.13.14	yes	384	43.13	even	21	inner