Embedded newform 43.6.g.a.10.13

• Label: 43.6.g.a.10.13
• Level: $43$
• Weight: $6$
• Character: 43.10
• Analytic conductor: $6.897$
• Analytic rank: $0$
• Dimension: $204$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.89650425196\)
Analytic rank:	\(0\)
Dimension:	\(204\)
Relative dimension:	\(17\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			10.13
Character	\(\chi\)	\(=\)	43.10
Dual form			43.6.g.a.13.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.33839 + 5.86386i) q^{2} +(10.9996 + 10.2062i) q^{3} +(-3.76256 + 1.81195i) q^{4} +(2.77137 - 0.417717i) q^{5} +(-45.1258 + 78.1601i) q^{6} +(82.5028 + 142.899i) q^{7} +(104.342 + 130.840i) q^{8} +(-1.33334 - 17.7922i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - 468 q^{4} - 85 q^{5} - 22 q^{6} + 454 q^{7} + 698 q^{8} - 333 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.33839	+	5.86386i	0.236596	+	1.03659i	0.944042	+	0.329826i	\(0.106990\pi\)
							−0.707446	+	0.706767i	\(0.750153\pi\)
\(3\)	10.9996	+	10.2062i	0.705627	+	0.654726i	0.949025	−	0.315200i	\(-0.102072\pi\)
							−0.243398	+	0.969926i	\(0.578262\pi\)
\(5\)	2.77137	−	0.417717i	0.0495757	−	0.00747234i	−0.124208	−	0.992256i	\(-0.539639\pi\)
							0.173783	+	0.984784i	\(0.444401\pi\)
\(7\)	82.5028	+	142.899i	0.636390	+	1.10226i	0.986219	+	0.165447i	\(0.0529066\pi\)
							−0.349828	+	0.936814i	\(0.613760\pi\)
\(11\)	−493.694	−	237.750i	−1.23020	−	0.592433i	−0.298065	−	0.954545i	\(-0.596341\pi\)
							−0.932134	+	0.362112i	\(0.882056\pi\)
\(13\)	51.7269	−	131.798i	0.0848902	−	0.216297i	−0.882125	−	0.471015i	\(-0.843888\pi\)
							0.967015	+	0.254719i	\(0.0819828\pi\)
\(17\)	200.408	+	30.2067i	0.168188	+	0.0253502i	0.232596	−	0.972573i	\(-0.425278\pi\)
							−0.0644086	+	0.997924i	\(0.520516\pi\)
\(19\)	19.8859	−	265.358i	0.0126375	−	0.168635i	−0.987314	−	0.158780i	\(-0.949244\pi\)
							0.999951	−	0.00985521i	\(-0.00313706\pi\)
\(23\)	1509.16	−	1028.92i	0.594860	−	0.405568i	−0.228125	−	0.973632i	\(-0.573260\pi\)
							0.822985	+	0.568063i	\(0.192307\pi\)
\(29\)	3957.85	−	3672.35i	0.873906	−	0.810866i	−0.109178	−	0.994022i	\(-0.534822\pi\)
							0.983084	+	0.183156i	\(0.0586314\pi\)
\(31\)	7685.82	+	2370.76i	1.43643	+	0.443081i	0.912640	−	0.408764i	\(-0.134040\pi\)
							0.523794	+	0.851845i	\(0.324516\pi\)
\(37\)	1583.76	−	2743.15i	0.190189	−	0.329417i	−0.755124	−	0.655582i	\(-0.772423\pi\)
							0.945313	+	0.326165i	\(0.105757\pi\)
\(41\)	−783.299	−	3431.86i	−0.0727726	−	0.318838i	0.925419	−	0.378945i	\(-0.123713\pi\)
							−0.998192	+	0.0601070i	\(0.980856\pi\)
\(43\)	−9750.05	+	7207.28i	−0.804148	+	0.594430i
							\(47\)	14474.8	−	6970.70i	0.955803	−	0.460290i	0.110086	−	0.993922i	\(-0.464887\pi\)
0.845717	+	0.533632i	\(0.179173\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.g.a.10.13	✓	204
43.13	even	21	inner	43.6.g.a.13.13	yes	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.g.a.10.13	✓	204	1.1	even	1	trivial
43.6.g.a.13.13	yes	204	43.13	even	21	inner