Embedded newform 43.6.g.a.9.11

• Label: 43.6.g.a.9.11
• Level: $43$
• Weight: $6$
• Character: 43.9
• 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			9.11
Character	\(\chi\)	\(=\)	43.9
Dual form			43.6.g.a.24.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.53765 + 3.18211i) q^{2} +(2.05098 + 0.309135i) q^{3} +(3.43451 - 15.0476i) q^{4} +(-53.1952 - 36.2679i) q^{5} +(4.22095 + 7.31090i) q^{6} +(95.1032 - 164.724i) q^{7} +(173.943 - 83.7664i) q^{8} +(-228.093 - 70.3575i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - 468 q^{4} - 85 q^{5} - 22 q^{6} + 454 q^{7} + 698 q^{8} - 333 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.53765	+	3.18211i	0.448597	+	0.562522i	0.953786	−	0.300486i	\(-0.0971489\pi\)
							−0.505190	+	0.863008i	\(0.668577\pi\)
\(3\)	2.05098	+	0.309135i	0.131570	+	0.0198310i	0.214497	−	0.976725i	\(-0.431189\pi\)
							−0.0829270	+	0.996556i	\(0.526427\pi\)
\(5\)	−53.1952	−	36.2679i	−0.951584	−	0.648779i	−0.0152481	−	0.999884i	\(-0.504854\pi\)
							−0.936336	+	0.351105i	\(0.885806\pi\)
\(7\)	95.1032	−	164.724i	0.733585	−	1.27061i	−0.221757	−	0.975102i	\(-0.571179\pi\)
							0.955342	−	0.295504i	\(-0.0954875\pi\)
\(11\)	96.9428	+	424.734i	0.241565	+	1.05837i	0.939593	+	0.342295i	\(0.111204\pi\)
							−0.698028	+	0.716071i	\(0.745939\pi\)
\(13\)	13.6916	−	182.702i	0.0224697	−	0.299837i	−0.974635	−	0.223799i	\(-0.928154\pi\)
							0.997105	−	0.0760377i	\(-0.0242269\pi\)
\(17\)	1594.33	−	1087.00i	1.33800	−	0.912233i	0.338414	−	0.940997i	\(-0.390110\pi\)
							0.999586	+	0.0287647i	\(0.00915734\pi\)
\(19\)	−580.594	+	179.089i	−0.368968	+	0.113811i	−0.473694	−	0.880690i	\(-0.657080\pi\)
							0.104726	+	0.994501i	\(0.466603\pi\)
\(23\)	−1037.96	+	963.085i	−0.409129	+	0.379616i	−0.857717	−	0.514123i	\(-0.828118\pi\)
							0.448588	+	0.893739i	\(0.351927\pi\)
\(29\)	2777.63	−	418.660i	0.613308	−	0.0924413i	0.164960	−	0.986300i	\(-0.447251\pi\)
							0.448348	+	0.893859i	\(0.352012\pi\)
\(31\)	−3014.63	+	7681.14i	−0.563416	+	1.43556i	0.311012	+	0.950406i	\(0.399332\pi\)
							−0.874428	+	0.485155i	\(0.838763\pi\)
\(37\)	−1498.58	−	2595.62i	−0.179960	−	0.311701i	0.761906	−	0.647687i	\(-0.224264\pi\)
							−0.941867	+	0.335987i	\(0.890930\pi\)
\(41\)	11660.1	+	14621.3i	1.08329	+	1.35840i	0.928875	+	0.370393i	\(0.120777\pi\)
							0.154413	+	0.988006i	\(0.450651\pi\)
\(43\)	9870.02	−	7042.10i	0.814042	−	0.580806i
							\(47\)	−491.637	+	2154.00i	−0.0324638	+	0.142233i	−0.988563	−	0.150811i	\(-0.951811\pi\)
0.956099	+	0.293045i	\(0.0946685\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.g.a.9.11	✓	204
43.24	even	21	inner	43.6.g.a.24.11	yes	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.g.a.9.11	✓	204	1.1	even	1	trivial
43.6.g.a.24.11	yes	204	43.24	even	21	inner