Embedded newform 201.4.i.b.22.13

• Label: 201.4.i.b.22.13
• 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			22.13
Character	\(\chi\)	\(=\)	201.22
Dual form			201.4.i.b.64.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.06465 - 0.899863i) q^{2} +(0.426945 + 2.96946i) q^{3} +(1.85231 - 1.19040i) q^{4} +(1.33631 - 2.92610i) q^{5} +(3.98055 + 8.71618i) q^{6} +(-27.3642 + 8.03485i) q^{7} +(-12.1277 + 13.9961i) q^{8} +(-8.63544 + 2.53559i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 170 q + 2 q^{2} + 51 q^{3} - 74 q^{4} - 4 q^{5} + 27 q^{6} + 12 q^{7} - 237 q^{8} - 153 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.06465	−	0.899863i	1.08352	−	0.318150i	0.309233	−	0.950986i	\(-0.399928\pi\)
							0.774285	+	0.632837i	\(0.218110\pi\)
\(3\)	0.426945	+	2.96946i	0.0821655	+	0.571474i
							\(5\)	1.33631	−	2.92610i	0.119523	−	0.261718i	−0.840409	−	0.541953i	\(-0.817685\pi\)
0.959932	+	0.280235i	\(0.0904123\pi\)
\(7\)	−27.3642	+	8.03485i	−1.47753	+	0.433841i	−0.918538	−	0.395333i	\(-0.870629\pi\)
							−0.558990	+	0.829175i	\(0.688811\pi\)
\(11\)	−11.6083	+	25.4187i	−0.318186	+	0.696729i	−0.999374	−	0.0353790i	\(-0.988736\pi\)
							0.681188	+	0.732108i	\(0.261463\pi\)
\(13\)	23.0926	+	26.6503i	0.492672	+	0.568574i	0.946578	−	0.322476i	\(-0.104515\pi\)
							−0.453905	+	0.891050i	\(0.649970\pi\)
\(17\)	−37.0176	−	23.7898i	−0.528123	−	0.339404i	0.249255	−	0.968438i	\(-0.419814\pi\)
							−0.777378	+	0.629034i	\(0.783451\pi\)
\(19\)	22.8375	+	6.70570i	0.275752	+	0.0809680i	0.416685	−	0.909051i	\(-0.363192\pi\)
							−0.140933	+	0.990019i	\(0.545010\pi\)
\(23\)	−10.5159	−	73.1398i	−0.0953357	−	0.663074i	−0.980314	−	0.197443i	\(-0.936736\pi\)
							0.884979	−	0.465631i	\(-0.154173\pi\)
\(29\)	35.4172			0.226786			0.113393	−	0.993550i	\(-0.463828\pi\)
							0.113393	+	0.993550i	\(0.463828\pi\)
\(31\)	−39.0307	+	45.0438i	−0.226133	+	0.260971i	−0.857466	−	0.514540i	\(-0.827963\pi\)
							0.631333	+	0.775512i	\(0.282508\pi\)
\(37\)	−150.682			−0.669511			−0.334755	−	0.942305i	\(-0.608654\pi\)
							−0.334755	+	0.942305i	\(0.608654\pi\)
\(41\)	−173.044	−	111.209i	−0.659144	−	0.423606i	0.167853	−	0.985812i	\(-0.446317\pi\)
							−0.826997	+	0.562206i	\(0.809953\pi\)
\(43\)	146.763	+	94.3189i	0.520492	+	0.334500i	0.774367	−	0.632737i	\(-0.218069\pi\)
							−0.253875	+	0.967237i	\(0.581705\pi\)
\(47\)	56.3993	+	392.266i	0.175036	+	1.21740i	0.868050	+	0.496477i	\(0.165373\pi\)
							−0.693014	+	0.720924i	\(0.743718\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.i.b.22.13	✓	170
67.64	even	11	inner	201.4.i.b.64.13	yes	170

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.i.b.22.13	✓	170	1.1	even	1	trivial
201.4.i.b.64.13	yes	170	67.64	even	11	inner