Embedded newform 201.4.f.a.38.8

• Label: 201.4.f.a.38.8
• Level: $201$
• Weight: $4$
• Character: 201.38
• Analytic conductor: $11.859$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	201.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8593839112\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(66\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			38.8
Character	\(\chi\)	\(=\)	201.38
Dual form			201.4.f.a.164.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.18859 + 3.79075i) q^{2} +(5.19172 - 0.214507i) q^{3} +(-5.57986 - 9.66459i) q^{4} -14.7915 q^{5} +(-10.5494 + 20.1500i) q^{6} +(21.3930 - 12.3512i) q^{7} +13.8306 q^{8} +(26.9080 - 2.22732i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 258 q^{4} - 23 q^{6} - 66 q^{7} - 70 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.18859	+	3.79075i	−0.773784	+	1.34023i	0.161692	+	0.986841i	\(0.448305\pi\)
							−0.935476	+	0.353391i	\(0.885028\pi\)
\(3\)	5.19172	−	0.214507i	0.999148	−	0.0412818i
							\(5\)	−14.7915			−1.32299			−0.661496	−	0.749949i	\(-0.730078\pi\)
−0.661496	+	0.749949i	\(0.730078\pi\)
\(7\)	21.3930	−	12.3512i	1.15511	−	0.666904i	0.204984	−	0.978765i	\(-0.434286\pi\)
							0.950128	+	0.311861i	\(0.100952\pi\)
\(11\)	−14.7782	−	25.5966i	−0.405072	−	0.701605i	0.589258	−	0.807945i	\(-0.299420\pi\)
							−0.994330	+	0.106340i	\(0.966087\pi\)
\(13\)	31.1199	+	17.9671i	0.663932	+	0.383322i	0.793774	−	0.608213i	\(-0.208113\pi\)
							−0.129841	+	0.991535i	\(0.541447\pi\)
\(17\)	88.2689	+	50.9621i	1.25932	+	0.727066i	0.972941	−	0.231052i	\(-0.0742168\pi\)
							0.286374	+	0.958118i	\(0.407550\pi\)
\(19\)	−8.95124	+	15.5040i	−0.108082	+	0.187203i	−0.914993	−	0.403469i	\(-0.867804\pi\)
							0.806911	+	0.590673i	\(0.201138\pi\)
\(23\)	5.46329	+	3.15423i	0.0495293	+	0.0285958i	0.524560	−	0.851373i	\(-0.324230\pi\)
							−0.475031	+	0.879969i	\(0.657563\pi\)
\(29\)	176.324	−	101.801i	1.12905	−	0.651858i	0.185355	−	0.982672i	\(-0.440657\pi\)
							0.943696	+	0.330814i	\(0.107323\pi\)
\(31\)	140.792	−	81.2865i	0.815711	−	0.470951i	−0.0332240	−	0.999448i	\(-0.510577\pi\)
							0.848935	+	0.528497i	\(0.177244\pi\)
\(37\)	8.27812	−	14.3381i	0.0367815	−	0.0637074i	−0.847049	−	0.531515i	\(-0.821623\pi\)
							0.883830	+	0.467808i	\(0.154956\pi\)
\(41\)	−5.19698	−	9.00143i	−0.0197959	−	0.0342875i	0.855958	−	0.517046i	\(-0.172968\pi\)
							−0.875754	+	0.482758i	\(0.839635\pi\)
\(43\)			275.543i			0.977207i	0.872506	+	0.488603i	\(0.162493\pi\)
							−0.872506	+	0.488603i	\(0.837507\pi\)
\(47\)	−451.643	+	260.756i	−1.40168	+	0.809259i	−0.994565	−	0.104118i	\(-0.966798\pi\)
							−0.407114	+	0.913378i	\(0.633465\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.f.a.38.8	✓	132
3.2	odd	2	inner	201.4.f.a.38.59	yes	132
67.30	odd	6	inner	201.4.f.a.164.59	yes	132
201.164	even	6	inner	201.4.f.a.164.8	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.f.a.38.8	✓	132	1.1	even	1	trivial
201.4.f.a.38.59	yes	132	3.2	odd	2	inner
201.4.f.a.164.8	yes	132	201.164	even	6	inner
201.4.f.a.164.59	yes	132	67.30	odd	6	inner