Embedded newform 201.4.i.a.22.4

• Label: 201.4.i.a.22.4
• 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			22.4
Character	\(\chi\)	\(=\)	201.22
Dual form			201.4.i.a.64.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.63496 + 1.06732i) q^{2} +(-0.426945 - 2.96946i) q^{3} +(5.34371 - 3.43419i) q^{4} +(-8.84178 + 19.3608i) q^{5} +(4.72129 + 10.3382i) q^{6} +(-29.4141 + 8.63677i) q^{7} +(4.08830 - 4.71815i) q^{8} +(-8.63544 + 2.53559i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 170 q + 2 q^{2} - 51 q^{3} - 82 q^{4} + 4 q^{5} - 27 q^{6} - 16 q^{7} - 177 q^{8} - 153 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.63496	+	1.06732i	−1.28515	+	0.377354i	−0.851798	−	0.523870i	\(-0.824488\pi\)
							−0.433353	+	0.901224i	\(0.642670\pi\)
\(3\)	−0.426945	−	2.96946i	−0.0821655	−	0.571474i
							\(5\)	−8.84178	+	19.3608i	−0.790833	+	1.73168i	−0.116581	+	0.993181i	\(0.537194\pi\)
−0.674252	+	0.738502i	\(0.735534\pi\)
\(7\)	−29.4141	+	8.63677i	−1.58821	+	0.466342i	−0.952235	−	0.305366i	\(-0.901221\pi\)
							−0.635978	+	0.771707i	\(0.719403\pi\)
\(11\)	−19.2918	+	42.2432i	−0.528792	+	1.15789i	0.437211	+	0.899359i	\(0.355966\pi\)
							−0.966002	+	0.258533i	\(0.916761\pi\)
\(13\)	17.1449	+	19.7863i	0.365780	+	0.422133i	0.908568	−	0.417737i	\(-0.137177\pi\)
							−0.542788	+	0.839870i	\(0.682631\pi\)
\(17\)	14.9360	+	9.59878i	0.213089	+	0.136944i	0.642832	−	0.766007i	\(-0.277759\pi\)
							−0.429743	+	0.902951i	\(0.641396\pi\)
\(19\)	−57.7568	−	16.9589i	−0.697385	−	0.204771i	−0.0862227	−	0.996276i	\(-0.527480\pi\)
							−0.611163	+	0.791505i	\(0.709298\pi\)
\(23\)	13.4647	+	93.6492i	0.122069	+	0.849009i	0.955206	+	0.295942i	\(0.0956336\pi\)
							−0.833137	+	0.553067i	\(0.813457\pi\)
\(29\)	81.9527			0.524767			0.262383	−	0.964964i	\(-0.415491\pi\)
							0.262383	+	0.964964i	\(0.415491\pi\)
\(31\)	26.9391	−	31.0894i	0.156078	−	0.180123i	−0.672325	−	0.740256i	\(-0.734704\pi\)
							0.828403	+	0.560133i	\(0.189250\pi\)
\(37\)	−283.987			−1.26182			−0.630908	−	0.775857i	\(-0.717318\pi\)
							−0.630908	+	0.775857i	\(0.717318\pi\)
\(41\)	179.437	+	115.317i	0.683495	+	0.439255i	0.835768	−	0.549083i	\(-0.185023\pi\)
							−0.152273	+	0.988338i	\(0.548659\pi\)
\(43\)	78.0981	+	50.1906i	0.276973	+	0.178000i	0.671752	−	0.740776i	\(-0.265542\pi\)
							−0.394778	+	0.918776i	\(0.629179\pi\)
\(47\)	−70.8785	−	492.971i	−0.219972	−	1.52994i	−0.738132	−	0.674657i	\(-0.764292\pi\)
							0.518160	−	0.855284i	\(-0.326617\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.i.a.22.4	✓	170
67.64	even	11	inner	201.4.i.a.64.4	yes	170

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.i.a.22.4	✓	170	1.1	even	1	trivial
201.4.i.a.64.4	yes	170	67.64	even	11	inner