Embedded newform 201.4.j.b.5.16

• Label: 201.4.j.b.5.16
• Level: $201$
• Weight: $4$
• Character: 201.5
• Analytic conductor: $11.859$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	201.j (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8593839112\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(64\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.16
Character	\(\chi\)	\(=\)	201.5
Dual form			201.4.j.b.161.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.08344 + 2.40442i) q^{2} +(4.78249 - 2.03169i) q^{3} +(-0.301982 - 2.10033i) q^{4} +(-1.53336 + 0.450234i) q^{5} +(-5.07900 + 15.7320i) q^{6} +(-6.46221 - 5.59953i) q^{7} +(-15.7323 - 10.1106i) q^{8} +(18.7445 - 19.4331i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 11 q^{3} - 290 q^{4} + 35 q^{6} - 22 q^{7} - 33 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.08344	+	2.40442i	−0.736607	+	0.850089i	−0.993199	−	0.116431i	\(-0.962855\pi\)
							0.256592	+	0.966520i	\(0.417400\pi\)
\(3\)	4.78249	−	2.03169i	0.920391	−	0.390999i
							\(5\)	−1.53336	+	0.450234i	−0.137147	+	0.0402701i	−0.349586	−	0.936904i	\(-0.613678\pi\)
0.212438	+	0.977174i	\(0.431860\pi\)
\(7\)	−6.46221	−	5.59953i	−0.348926	−	0.302346i	0.462710	−	0.886510i	\(-0.346877\pi\)
							−0.811636	+	0.584163i	\(0.801423\pi\)
\(11\)	−30.7539	+	9.03015i	−0.842967	+	0.247517i	−0.674578	−	0.738203i	\(-0.735675\pi\)
							−0.168389	+	0.985721i	\(0.553856\pi\)
\(13\)	−28.2349	−	43.9344i	−0.602382	−	0.937325i	−0.999806	−	0.0196896i	\(-0.993732\pi\)
							0.397424	−	0.917635i	\(-0.369904\pi\)
\(17\)	−9.15186	−	1.31584i	−0.130568	−	0.0187728i	0.0767210	−	0.997053i	\(-0.475555\pi\)
							−0.207289	+	0.978280i	\(0.566464\pi\)
\(19\)	−30.6034	−	35.3182i	−0.369521	−	0.426450i	0.540286	−	0.841481i	\(-0.318316\pi\)
							−0.909807	+	0.415031i	\(0.863771\pi\)
\(23\)	−113.111	+	51.6560i	−1.02544	+	0.468305i	−0.855858	−	0.517210i	\(-0.826971\pi\)
							−0.169587	+	0.985515i	\(0.554243\pi\)
\(29\)		−	250.085i		−	1.60137i	−0.599089	−	0.800683i	\(-0.704470\pi\)
							0.599089	−	0.800683i	\(-0.295530\pi\)
\(31\)	−103.053	+	160.353i	−0.597059	+	0.929042i	0.402846	+	0.915268i	\(0.368021\pi\)
							−0.999905	+	0.0137744i	\(0.995615\pi\)
\(37\)	275.770			1.22531			0.612653	−	0.790352i	\(-0.290102\pi\)
							0.612653	+	0.790352i	\(0.290102\pi\)
\(41\)	42.7504	−	297.336i	0.162841	−	1.13259i	−0.730404	−	0.683016i	\(-0.760668\pi\)
							0.893245	−	0.449570i	\(-0.148423\pi\)
\(43\)	−170.204	−	24.4717i	−0.603627	−	0.0867884i	−0.166276	−	0.986079i	\(-0.553174\pi\)
							−0.437351	+	0.899291i	\(0.644083\pi\)
\(47\)	394.990	−	180.386i	1.22585	−	0.559829i	0.305978	−	0.952038i	\(-0.401016\pi\)
							0.919876	+	0.392210i	\(0.128289\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.j.b.5.16	✓	640
3.2	odd	2	inner	201.4.j.b.5.49	yes	640
67.27	odd	22	inner	201.4.j.b.161.49	yes	640
201.161	even	22	inner	201.4.j.b.161.16	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.j.b.5.16	✓	640	1.1	even	1	trivial
201.4.j.b.5.49	yes	640	3.2	odd	2	inner
201.4.j.b.161.16	yes	640	201.161	even	22	inner
201.4.j.b.161.49	yes	640	67.27	odd	22	inner