Embedded newform 201.4.i.a.22.7

• Label: 201.4.i.a.22.7
• Level: $201$
• Weight: $4$
• Character: 201.22
• Analytic conductor: $11.859$
• Analytic rank: $0$
• Dimension: $170$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	201.i (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8593839112\)
Analytic rank:	\(0\)
Dimension:	\(170\)
Relative dimension:	\(17\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			22.7
Character	\(\chi\)	\(=\)	201.22
Dual form			201.4.i.a.64.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.71481 + 0.503514i) q^{2} +(-0.426945 - 2.96946i) q^{3} +(-4.04298 + 2.59827i) q^{4} +(-2.04662 + 4.48147i) q^{5} +(2.22729 + 4.87710i) q^{6} +(13.3024 - 3.90593i) q^{7} +(14.9876 - 17.2967i) q^{8} +(-8.63544 + 2.53559i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 170 q + 2 q^{2} - 51 q^{3} - 82 q^{4} + 4 q^{5} - 27 q^{6} - 16 q^{7} - 177 q^{8} - 153 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{10}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.71481	+	0.503514i	−0.606277	+	0.178019i	−0.570441	−	0.821339i	\(-0.693228\pi\)
							−0.0358361	+	0.999358i	\(0.511409\pi\)
\(3\)	−0.426945	−	2.96946i	−0.0821655	−	0.571474i
							\(5\)	−2.04662	+	4.48147i	−0.183055	+	0.400835i	−0.978806	−	0.204789i	\(-0.934349\pi\)
0.795751	+	0.605624i	\(0.207076\pi\)
\(7\)	13.3024	−	3.90593i	0.718261	−	0.210900i	0.0978766	−	0.995199i	\(-0.468795\pi\)
							0.620384	+	0.784298i	\(0.286977\pi\)
\(11\)	1.31674	−	2.88326i	0.0360920	−	0.0790305i	−0.890727	−	0.454538i	\(-0.849804\pi\)
							0.926819	+	0.375508i	\(0.122532\pi\)
\(13\)	25.3767	+	29.2862i	0.541402	+	0.624811i	0.958858	−	0.283886i	\(-0.0916238\pi\)
							−0.417456	+	0.908697i	\(0.637078\pi\)
\(17\)	−40.3325	−	25.9201i	−0.575416	−	0.369798i	0.220332	−	0.975425i	\(-0.429286\pi\)
							−0.795748	+	0.605627i	\(0.792922\pi\)
\(19\)	−145.741	−	42.7933i	−1.75975	−	0.516708i	−0.767505	−	0.641042i	\(-0.778502\pi\)
							−0.992240	+	0.124334i	\(0.960320\pi\)
\(23\)	7.30055	+	50.7764i	0.0661856	+	0.460331i	0.995782	+	0.0917517i	\(0.0292466\pi\)
							−0.929596	+	0.368579i	\(0.879844\pi\)
\(29\)	−148.501			−0.950893			−0.475446	−	0.879745i	\(-0.657713\pi\)
							−0.475446	+	0.879745i	\(0.657713\pi\)
\(31\)	−202.806	+	234.051i	−1.17500	+	1.35602i	−0.253648	+	0.967296i	\(0.581631\pi\)
							−0.921353	+	0.388727i	\(0.872915\pi\)
\(37\)	−141.618			−0.629239			−0.314619	−	0.949218i	\(-0.601877\pi\)
							−0.314619	+	0.949218i	\(0.601877\pi\)
\(41\)	−415.055	−	266.740i	−1.58099	−	1.01604i	−0.975447	−	0.220233i	\(-0.929318\pi\)
							−0.605546	−	0.795810i	\(-0.707045\pi\)
\(43\)	311.626	+	200.270i	1.10518	+	0.710254i	0.960237	−	0.279187i	\(-0.0900649\pi\)
							0.144940	+	0.989441i	\(0.453701\pi\)
\(47\)	−76.4749	−	531.895i	−0.237341	−	1.65074i	−0.665032	−	0.746815i	\(-0.731582\pi\)
							0.427691	−	0.903925i	\(-0.359327\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.i.a.22.7	✓	170
67.64	even	11	inner	201.4.i.a.64.7	yes	170

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.i.a.22.7	✓	170	1.1	even	1	trivial
201.4.i.a.64.7	yes	170	67.64	even	11	inner