Embedded newform 201.4.f.a.38.9

• Label: 201.4.f.a.38.9
• 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			38.9
Character	\(\chi\)	\(=\)	201.38
Dual form			201.4.f.a.164.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.17777 + 3.77200i) q^{2} +(-5.01884 - 1.34584i) q^{3} +(-5.48532 - 9.50086i) q^{4} +15.8201 q^{5} +(16.0064 - 16.0001i) q^{6} +(-22.0161 + 12.7110i) q^{7} +12.9387 q^{8} +(23.3774 + 13.5091i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 258 q^{4} - 23 q^{6} - 66 q^{7} - 70 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.17777	+	3.77200i	−0.769956	+	1.33360i	0.167630	+	0.985850i	\(0.446389\pi\)
							−0.937586	+	0.347753i	\(0.886945\pi\)
\(3\)	−5.01884	−	1.34584i	−0.965875	−	0.259007i
							\(5\)	15.8201			1.41499			0.707497	−	0.706716i	\(-0.249824\pi\)
0.707497	+	0.706716i	\(0.249824\pi\)
\(7\)	−22.0161	+	12.7110i	−1.18876	+	0.686329i	−0.958024	−	0.286688i	\(-0.907446\pi\)
							−0.230733	+	0.973017i	\(0.574112\pi\)
\(11\)	27.9178	+	48.3551i	0.765231	+	1.32542i	0.940124	+	0.340832i	\(0.110709\pi\)
							−0.174893	+	0.984587i	\(0.555958\pi\)
\(13\)	−62.1629	−	35.8898i	−1.32622	−	0.765695i	−0.341509	−	0.939878i	\(-0.610938\pi\)
							−0.984713	+	0.174183i	\(0.944271\pi\)
\(17\)	−82.8000	−	47.8046i	−1.18129	−	0.682019i	−0.224978	−	0.974364i	\(-0.572231\pi\)
							−0.956313	+	0.292345i	\(0.905564\pi\)
\(19\)	−15.9505	+	27.6271i	−0.192594	+	0.333583i	−0.946109	−	0.323848i	\(-0.895023\pi\)
							0.753515	+	0.657431i	\(0.228357\pi\)
\(23\)	79.1678	+	45.7076i	0.717723	+	0.414378i	0.813914	−	0.580985i	\(-0.197333\pi\)
							−0.0961909	+	0.995363i	\(0.530666\pi\)
\(29\)	−138.369	+	79.8874i	−0.886017	+	0.511542i	−0.872637	−	0.488369i	\(-0.837592\pi\)
							−0.0133791	+	0.999910i	\(0.504259\pi\)
\(31\)	165.435	−	95.5141i	0.958485	−	0.553382i	0.0627787	−	0.998027i	\(-0.480004\pi\)
							0.895706	+	0.444646i	\(0.146670\pi\)
\(37\)	23.0188	−	39.8696i	0.102277	−	0.177149i	−0.810345	−	0.585953i	\(-0.800720\pi\)
							0.912623	+	0.408803i	\(0.134054\pi\)
\(41\)	−259.685	−	449.788i	−0.989171	−	1.71329i	−0.621689	−	0.783264i	\(-0.713553\pi\)
							−0.367482	−	0.930031i	\(-0.619780\pi\)
\(43\)		−	525.888i		−	1.86505i	−0.361107	−	0.932525i	\(-0.617601\pi\)
							0.361107	−	0.932525i	\(-0.382399\pi\)
\(47\)	−38.0237	+	21.9530i	−0.118007	+	0.0681314i	−0.557842	−	0.829947i	\(-0.688370\pi\)
							0.439835	+	0.898079i	\(0.355037\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.f.a.38.9	✓	132
3.2	odd	2	inner	201.4.f.a.38.58	yes	132
67.30	odd	6	inner	201.4.f.a.164.58	yes	132
201.164	even	6	inner	201.4.f.a.164.9	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.f.a.38.9	✓	132	1.1	even	1	trivial
201.4.f.a.38.58	yes	132	3.2	odd	2	inner
201.4.f.a.164.9	yes	132	201.164	even	6	inner
201.4.f.a.164.58	yes	132	67.30	odd	6	inner