Embedded newform 201.4.i.a.22.14

• Label: 201.4.i.a.22.14
• 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			22.14
Character	\(\chi\)	\(=\)	201.22
Dual form			201.4.i.a.64.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.31194 - 0.972475i) q^{2} +(-0.426945 - 2.96946i) q^{3} +(3.29324 - 2.11644i) q^{4} +(-5.88287 + 12.8817i) q^{5} +(-4.30175 - 9.41951i) q^{6} +(-7.18856 + 2.11075i) q^{7} +(-9.23455 + 10.6572i) q^{8} +(-8.63544 + 2.53559i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 170 q + 2 q^{2} - 51 q^{3} - 82 q^{4} + 4 q^{5} - 27 q^{6} - 16 q^{7} - 177 q^{8} - 153 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.31194	−	0.972475i	1.17095	−	0.343822i	0.362271	−	0.932073i	\(-0.382002\pi\)
							0.808678	+	0.588251i	\(0.200183\pi\)
\(3\)	−0.426945	−	2.96946i	−0.0821655	−	0.571474i
							\(5\)	−5.88287	+	12.8817i	−0.526180	+	1.15217i	0.440866	+	0.897573i	\(0.354671\pi\)
−0.967046	+	0.254601i	\(0.918056\pi\)
\(7\)	−7.18856	+	2.11075i	−0.388146	+	0.113970i	−0.469982	−	0.882676i	\(-0.655740\pi\)
							0.0818365	+	0.996646i	\(0.473921\pi\)
\(11\)	−11.6755	+	25.5658i	−0.320027	+	0.700761i	−0.999456	−	0.0329802i	\(-0.989500\pi\)
							0.679429	+	0.733741i	\(0.262227\pi\)
\(13\)	30.6863	+	35.4139i	0.654681	+	0.755542i	0.981899	−	0.189407i	\(-0.0606564\pi\)
							−0.327217	+	0.944949i	\(0.606111\pi\)
\(17\)	−6.27485	−	4.03260i	−0.0895221	−	0.0575324i	0.495114	−	0.868828i	\(-0.335126\pi\)
							−0.584636	+	0.811296i	\(0.698763\pi\)
\(19\)	8.93529	+	2.62364i	0.107889	+	0.0316791i	0.335232	−	0.942136i	\(-0.391185\pi\)
							−0.227342	+	0.973815i	\(0.573004\pi\)
\(23\)	−18.6019	−	129.379i	−0.168642	−	1.17293i	−0.881694	−	0.471821i	\(-0.843597\pi\)
							0.713052	−	0.701111i	\(-0.247312\pi\)
\(29\)	−187.485			−1.20052			−0.600259	−	0.799806i	\(-0.704936\pi\)
							−0.600259	+	0.799806i	\(0.704936\pi\)
\(31\)	−71.1144	+	82.0704i	−0.412017	+	0.475493i	−0.923389	−	0.383867i	\(-0.874592\pi\)
							0.511371	+	0.859360i	\(0.329138\pi\)
\(37\)	259.976			1.15513			0.577566	−	0.816344i	\(-0.304003\pi\)
							0.577566	+	0.816344i	\(0.304003\pi\)
\(41\)	180.188	+	115.800i	0.686357	+	0.441095i	0.836789	−	0.547525i	\(-0.184430\pi\)
							−0.150432	+	0.988620i	\(0.548066\pi\)
\(43\)	92.9914	+	59.7619i	0.329792	+	0.211944i	0.695049	−	0.718962i	\(-0.255383\pi\)
							−0.365257	+	0.930907i	\(0.619019\pi\)
\(47\)	23.2484	+	161.696i	0.0721516	+	0.501825i	0.993567	+	0.113246i	\(0.0361249\pi\)
							−0.921415	+	0.388579i	\(0.872966\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.i.a.22.14	✓	170
67.64	even	11	inner	201.4.i.a.64.14	yes	170

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.i.a.22.14	✓	170	1.1	even	1	trivial
201.4.i.a.64.14	yes	170	67.64	even	11	inner