Embedded newform 201.2.j.a.5.14

• Label: 201.2.j.a.5.14
• Level: $201$
• Weight: $2$
• Character: 201.5
• Analytic conductor: $1.605$
• Analytic rank: $0$
• Dimension: $200$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	201.j (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.60499308063\)
Analytic rank:	\(0\)
Dimension:	\(200\)
Relative dimension:	\(20\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.14
Character	\(\chi\)	\(=\)	201.5
Dual form			201.2.j.a.161.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.634589 - 0.732355i) q^{2} +(-0.189712 - 1.72163i) q^{3} +(0.150989 + 1.05015i) q^{4} +(2.77562 - 0.814995i) q^{5} +(-1.38123 - 0.953590i) q^{6} +(0.0640238 + 0.0554769i) q^{7} +(2.49533 + 1.60365i) q^{8} +(-2.92802 + 0.653229i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 11 q^{3} - 34 q^{4} - 7 q^{6} - 22 q^{7} + 3 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.634589	−	0.732355i	0.448722	−	0.517853i	−0.485649	−	0.874154i	\(-0.661417\pi\)
							0.934371	+	0.356301i	\(0.115962\pi\)
\(3\)	−0.189712	−	1.72163i	−0.109530	−	0.993983i
							\(5\)	2.77562	−	0.814995i	1.24129	−	0.364477i	0.405793	−	0.913965i	\(-0.366995\pi\)
0.835501	+	0.549488i	\(0.185177\pi\)
\(7\)	0.0640238	+	0.0554769i	0.0241987	+	0.0209683i	0.666876	−	0.745169i	\(-0.267631\pi\)
							−0.642677	+	0.766137i	\(0.722176\pi\)
\(11\)	−0.568757	+	0.167002i	−0.171487	+	0.0503531i	−0.366349	−	0.930478i	\(-0.619392\pi\)
							0.194862	+	0.980831i	\(0.437574\pi\)
\(13\)	−1.65885	−	2.58123i	−0.460083	−	0.715903i	0.531259	−	0.847210i	\(-0.321719\pi\)
							−0.991342	+	0.131306i	\(0.958083\pi\)
\(17\)	−7.05609	−	1.01451i	−1.71135	−	0.246055i	−0.784139	−	0.620586i	\(-0.786895\pi\)
							−0.927215	+	0.374530i	\(0.877804\pi\)
\(19\)	3.33775	+	3.85197i	0.765733	+	0.883703i	0.995993	−	0.0894263i	\(-0.0285034\pi\)
							−0.230261	+	0.973129i	\(0.573958\pi\)
\(23\)	2.13181	−	0.973563i	0.444512	−	0.203002i	−0.180568	−	0.983563i	\(-0.557794\pi\)
							0.625080	+	0.780561i	\(0.285066\pi\)
\(29\)		−	5.14027i		−	0.954524i	−0.878761	−	0.477262i	\(-0.841629\pi\)
							0.878761	−	0.477262i	\(-0.158371\pi\)
\(31\)	−5.23068	+	8.13910i	−0.939459	+	1.46183i	−0.0532278	+	0.998582i	\(0.516951\pi\)
							−0.886231	+	0.463244i	\(0.846685\pi\)
\(37\)	3.85844			0.634324			0.317162	−	0.948371i	\(-0.397270\pi\)
							0.317162	+	0.948371i	\(0.397270\pi\)
\(41\)	−0.976063	+	6.78867i	−0.152435	+	1.06021i	0.759685	+	0.650291i	\(0.225353\pi\)
							−0.912121	+	0.409921i	\(0.865556\pi\)
\(43\)	10.3247	+	1.48447i	1.57450	+	0.226379i	0.873516	−	0.486796i	\(-0.161834\pi\)
							0.700986	+	0.713175i	\(0.252743\pi\)
\(47\)	−3.30392	+	1.50885i	−0.481926	+	0.220088i	−0.641537	−	0.767092i	\(-0.721703\pi\)
							0.159612	+	0.987180i	\(0.448976\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.j.a.5.14	yes	200
3.2	odd	2	inner	201.2.j.a.5.7	✓	200
67.27	odd	22	inner	201.2.j.a.161.7	yes	200
201.161	even	22	inner	201.2.j.a.161.14	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.j.a.5.7	✓	200	3.2	odd	2	inner
201.2.j.a.5.14	yes	200	1.1	even	1	trivial
201.2.j.a.161.7	yes	200	67.27	odd	22	inner
201.2.j.a.161.14	yes	200	201.161	even	22	inner