Embedded newform 201.2.j.a.5.15

• Label: 201.2.j.a.5.15
• Level: $201$
• Weight: $2$
• Character: 201.5
• Analytic conductor: $1.605$
• Analytic rank: $0$
• Dimension: $200$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	201.j (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.60499308063\)
Analytic rank:	\(0\)
Dimension:	\(200\)
Relative dimension:	\(20\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.15
Character	\(\chi\)	\(=\)	201.5
Dual form			201.2.j.a.161.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.706711 - 0.815588i) q^{2} +(-0.826962 + 1.52189i) q^{3} +(0.118886 + 0.826873i) q^{4} +(-3.89728 + 1.14434i) q^{5} +(0.656808 + 1.74999i) q^{6} +(-1.64135 - 1.42224i) q^{7} +(2.57413 + 1.65429i) q^{8} +(-1.63227 - 2.51708i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 11 q^{3} - 34 q^{4} - 7 q^{6} - 22 q^{7} + 3 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.706711	−	0.815588i	0.499720	−	0.576708i	−0.448717	−	0.893674i	\(-0.648119\pi\)
							0.948437	+	0.316966i	\(0.102664\pi\)
\(3\)	−0.826962	+	1.52189i	−0.477447	+	0.878661i
							\(5\)	−3.89728	+	1.14434i	−1.74292	+	0.511766i	−0.989344	−	0.145598i	\(-0.953489\pi\)
−0.753573	+	0.657365i	\(0.771671\pi\)
\(7\)	−1.64135	−	1.42224i	−0.620373	−	0.537556i	0.286974	−	0.957938i	\(-0.407351\pi\)
							−0.907347	+	0.420382i	\(0.861896\pi\)
\(11\)	−2.88196	+	0.846220i	−0.868944	+	0.255145i	−0.685666	−	0.727916i	\(-0.740489\pi\)
							−0.183278	+	0.983061i	\(0.558671\pi\)
\(13\)	3.30823	+	5.14771i	0.917538	+	1.42772i	0.903869	+	0.427810i	\(0.140715\pi\)
							0.0136692	+	0.999907i	\(0.495649\pi\)
\(17\)	−0.695221	−	0.0999577i	−0.168616	−	0.0242433i	0.0574898	−	0.998346i	\(-0.481690\pi\)
							−0.226106	+	0.974103i	\(0.572599\pi\)
\(19\)	2.90773	+	3.35570i	0.667079	+	0.769850i	0.983916	−	0.178629i	\(-0.0571663\pi\)
							−0.316837	+	0.948480i	\(0.602621\pi\)
\(23\)	2.23010	−	1.01845i	0.465008	−	0.212362i	−0.169112	−	0.985597i	\(-0.554090\pi\)
							0.634120	+	0.773235i	\(0.281363\pi\)
\(29\)			0.919219i			0.170695i	0.996351	+	0.0853473i	\(0.0272000\pi\)
							−0.996351	+	0.0853473i	\(0.972800\pi\)
\(31\)	0.288341	−	0.448667i	0.0517876	−	0.0805830i	−0.814394	−	0.580312i	\(-0.802931\pi\)
							0.866182	+	0.499729i	\(0.166567\pi\)
\(37\)	−11.1226			−1.82854			−0.914269	−	0.405107i	\(-0.867234\pi\)
							−0.914269	+	0.405107i	\(0.867234\pi\)
\(41\)	−0.950449	+	6.61052i	−0.148435	+	1.03239i	0.770347	+	0.637625i	\(0.220083\pi\)
							−0.918782	+	0.394765i	\(0.870826\pi\)
\(43\)	5.20413	+	0.748241i	0.793622	+	0.114106i	0.527189	−	0.849748i	\(-0.323246\pi\)
							0.266433	+	0.963853i	\(0.414155\pi\)
\(47\)	−0.838249	+	0.382815i	−0.122271	+	0.0558394i	−0.475611	−	0.879656i	\(-0.657773\pi\)
							0.353340	+	0.935495i	\(0.385046\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.j.a.5.15	yes	200
3.2	odd	2	inner	201.2.j.a.5.6	✓	200
67.27	odd	22	inner	201.2.j.a.161.6	yes	200
201.161	even	22	inner	201.2.j.a.161.15	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.j.a.5.6	✓	200	3.2	odd	2	inner
201.2.j.a.5.15	yes	200	1.1	even	1	trivial
201.2.j.a.161.6	yes	200	67.27	odd	22	inner
201.2.j.a.161.15	yes	200	201.161	even	22	inner