Embedded newform 201.2.j.a.5.1

• Label: 201.2.j.a.5.1
• 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			5.1
Character	\(\chi\)	\(=\)	201.5
Dual form			201.2.j.a.161.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.80413 + 2.08208i) q^{2} +(-1.15463 - 1.29106i) q^{3} +(-0.795528 - 5.53302i) q^{4} +(-0.470328 + 0.138101i) q^{5} +(4.77118 - 0.0747868i) q^{6} +(-0.367215 - 0.318193i) q^{7} +(8.32012 + 5.34702i) q^{8} +(-0.333667 + 2.98139i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 11 q^{3} - 34 q^{4} - 7 q^{6} - 22 q^{7} + 3 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.80413	+	2.08208i	−1.27571	+	1.47225i	−0.466748	+	0.884390i	\(0.654575\pi\)
							−0.808963	+	0.587859i	\(0.799971\pi\)
\(3\)	−1.15463	−	1.29106i	−0.666625	−	0.745393i
							\(5\)	−0.470328	+	0.138101i	−0.210337	+	0.0617605i	−0.385204	−	0.922832i	\(-0.625869\pi\)
0.174867	+	0.984592i	\(0.444051\pi\)
\(7\)	−0.367215	−	0.318193i	−0.138794	−	0.120266i	0.582697	−	0.812690i	\(-0.301997\pi\)
							−0.721491	+	0.692424i	\(0.756543\pi\)
\(11\)	−2.75940	+	0.810233i	−0.831990	+	0.244294i	−0.669872	−	0.742477i	\(-0.733651\pi\)
							−0.162119	+	0.986771i	\(0.551833\pi\)
\(13\)	3.22653	+	5.02058i	0.894879	+	1.39246i	0.919639	+	0.392765i	\(0.128481\pi\)
							−0.0247600	+	0.999693i	\(0.507882\pi\)
\(17\)	1.94778	+	0.280048i	0.472406	+	0.0679217i	0.374406	−	0.927265i	\(-0.377847\pi\)
							0.0979995	+	0.995186i	\(0.468756\pi\)
\(19\)	1.54714	+	1.78549i	0.354938	+	0.409620i	0.904937	−	0.425545i	\(-0.139917\pi\)
							−0.549999	+	0.835165i	\(0.685372\pi\)
\(23\)	4.40280	−	2.01069i	0.918048	−	0.419259i	0.100380	−	0.994949i	\(-0.467994\pi\)
							0.817668	+	0.575691i	\(0.195267\pi\)
\(29\)			2.62917i			0.488225i	0.969747	+	0.244113i	\(0.0784967\pi\)
							−0.969747	+	0.244113i	\(0.921503\pi\)
\(31\)	−4.28104	+	6.66143i	−0.768898	+	1.19643i	0.207030	+	0.978335i	\(0.433620\pi\)
							−0.975928	+	0.218094i	\(0.930016\pi\)
\(37\)	6.40071			1.05227			0.526135	−	0.850401i	\(-0.323640\pi\)
							0.526135	+	0.850401i	\(0.323640\pi\)
\(41\)	−1.37816	+	9.58533i	−0.215233	+	1.49698i	0.540080	+	0.841614i	\(0.318394\pi\)
							−0.755313	+	0.655364i	\(0.772515\pi\)
\(43\)	2.89581	+	0.416354i	0.441607	+	0.0634934i	0.359532	−	0.933133i	\(-0.382936\pi\)
							0.0820746	+	0.996626i	\(0.473845\pi\)
\(47\)	−4.00725	+	1.83005i	−0.584518	+	0.266940i	−0.685648	−	0.727933i	\(-0.740481\pi\)
							0.101131	+	0.994873i	\(0.467754\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.j.a.5.1	✓	200
3.2	odd	2	inner	201.2.j.a.5.20	yes	200
67.27	odd	22	inner	201.2.j.a.161.20	yes	200
201.161	even	22	inner	201.2.j.a.161.1	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.j.a.5.1	✓	200	1.1	even	1	trivial
201.2.j.a.5.20	yes	200	3.2	odd	2	inner
201.2.j.a.161.1	yes	200	201.161	even	22	inner
201.2.j.a.161.20	yes	200	67.27	odd	22	inner