Embedded newform 43.8.g.a.9.14

• Label: 43.8.g.a.9.14
• Level: $43$
• Weight: $8$
• Character: 43.9
• Analytic conductor: $13.433$
• Analytic rank: $0$
• Dimension: $300$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.4325560958\)
Analytic rank:	\(0\)
Dimension:	\(300\)
Relative dimension:	\(25\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			9.14
Character	\(\chi\)	\(=\)	43.9
Dual form			43.8.g.a.24.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.22112 + 2.78519i) q^{2} +(-37.3010 - 5.62222i) q^{3} +(25.6587 - 112.418i) q^{4} +(39.7972 + 27.1333i) q^{5} +(-67.1910 - 116.378i) q^{6} +(-76.5793 + 132.639i) q^{7} +(780.928 - 376.075i) q^{8} +(-730.081 - 225.200i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 40 q^{2} - 12 q^{3} - 3028 q^{4} + 235 q^{5} + 1274 q^{6} - 4122 q^{7} + 762 q^{8} + 8097 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.22112	+	2.78519i	0.196321	+	0.246179i	0.870242	−	0.492625i	\(-0.163963\pi\)
							−0.673921	+	0.738804i	\(0.735391\pi\)
\(3\)	−37.3010	−	5.62222i	−0.797620	−	0.120222i	−0.262427	−	0.964952i	\(-0.584523\pi\)
							−0.535193	+	0.844730i	\(0.679761\pi\)
\(5\)	39.7972	+	27.1333i	0.142383	+	0.0970749i	0.632416	−	0.774629i	\(-0.282063\pi\)
							−0.490034	+	0.871703i	\(0.663016\pi\)
\(7\)	−76.5793	+	132.639i	−0.0843855	+	0.146160i	−0.905129	−	0.425137i	\(-0.860226\pi\)
							0.820744	+	0.571297i	\(0.193559\pi\)
\(11\)	247.552	+	1084.60i	0.0560779	+	0.245694i	0.995196	−	0.0979006i	\(-0.0312127\pi\)
							−0.939118	+	0.343594i	\(0.888356\pi\)
\(13\)	−542.368	+	7237.40i	−0.0684688	+	0.913653i	0.852177	+	0.523254i	\(0.175282\pi\)
							−0.920646	+	0.390399i	\(0.872337\pi\)
\(17\)	−23860.6	+	16267.9i	−1.17790	+	0.803082i	−0.984040	−	0.177947i	\(-0.943055\pi\)
							−0.193864	+	0.981028i	\(0.562102\pi\)
\(19\)	−33347.5	+	10286.3i	−1.11539	+	0.344051i	−0.797042	−	0.603924i	\(-0.793603\pi\)
							−0.318344	+	0.947975i	\(0.603127\pi\)
\(23\)	−8253.57	+	7658.19i	−0.141447	+	0.131244i	−0.747723	−	0.664010i	\(-0.768853\pi\)
							0.606276	+	0.795254i	\(0.292663\pi\)
\(29\)	−15211.5	+	2292.76i	−0.115819	+	0.0174568i	−0.206696	−	0.978405i	\(-0.566271\pi\)
							0.0908772	+	0.995862i	\(0.471033\pi\)
\(31\)	−102376.	+	260849.i	−0.617207	+	1.57262i	0.189639	+	0.981854i	\(0.439268\pi\)
							−0.806845	+	0.590763i	\(0.798827\pi\)
\(37\)	100741.	+	174489.i	0.326965	+	0.566320i	0.981908	−	0.189358i	\(-0.0606407\pi\)
							−0.654943	+	0.755678i	\(0.727307\pi\)
\(41\)	140957.	+	176755.i	0.319406	+	0.400523i	0.915452	−	0.402428i	\(-0.131834\pi\)
							−0.596045	+	0.802951i	\(0.703262\pi\)
\(43\)	209663.	−	477347.i	0.402144	−	0.915576i
							\(47\)	55359.6	−	242546.i	0.0777769	−	0.340763i	−0.921036	−	0.389478i	\(-0.872656\pi\)
0.998813	+	0.0487151i	\(0.0155126\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.g.a.9.14	✓	300
43.24	even	21	inner	43.8.g.a.24.14	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.g.a.9.14	✓	300	1.1	even	1	trivial
43.8.g.a.24.14	yes	300	43.24	even	21	inner