Embedded newform 200.2.o.a.69.7

• Label: 200.2.o.a.69.7
• Level: $200$
• Weight: $2$
• Character: 200.69
• Analytic conductor: $1.597$
• Analytic rank: $0$
• Dimension: $112$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	200.o (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.59700804043\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(28\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			69.7
Character	\(\chi\)	\(=\)	200.69
Dual form			200.2.o.a.29.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08090 - 0.911952i) q^{2} +(2.58107 - 1.87526i) q^{3} +(0.336688 + 1.97146i) q^{4} +(0.602254 - 2.15344i) q^{5} +(-4.50002 - 0.326846i) q^{6} +2.97769i q^{7} +(1.43395 - 2.43799i) q^{8} +(2.21828 - 6.82716i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 5 q^{2} - 3 q^{4} + q^{6} + 10 q^{8} - 30 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(151\)	\(177\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08090	−	0.911952i	−0.764312	−	0.644847i
							\(3\)	2.58107	−	1.87526i	1.49018	−	1.08268i	0.516086	−	0.856537i	\(-0.327388\pi\)
0.974094	−	0.226143i	\(-0.0726116\pi\)
\(5\)	0.602254	−	2.15344i	0.269336	−	0.963046i
							\(7\)			2.97769i			1.12546i	0.826641	+	0.562730i	\(0.190249\pi\)
−0.826641	+	0.562730i	\(0.809751\pi\)
\(11\)	0.263378	−	0.0855768i	0.0794116	−	0.0258024i	−0.269042	−	0.963129i	\(-0.586707\pi\)
							0.348453	+	0.937326i	\(0.386707\pi\)
\(13\)	−1.36304	+	4.19500i	−0.378039	+	1.16349i	0.563366	+	0.826207i	\(0.309506\pi\)
							−0.941405	+	0.337278i	\(0.890494\pi\)
\(17\)	−3.01210	+	4.14580i	−0.730542	+	1.00551i	0.268565	+	0.963262i	\(0.413451\pi\)
							−0.999107	+	0.0422436i	\(0.986549\pi\)
\(19\)	0.824879	−	1.13535i	0.189240	−	0.260467i	−0.703846	−	0.710353i	\(-0.748535\pi\)
							0.893086	+	0.449886i	\(0.148535\pi\)
\(23\)	−0.307187	+	0.0998113i	−0.0640530	+	0.0208121i	−0.340868	−	0.940111i	\(-0.610721\pi\)
							0.276815	+	0.960923i	\(0.410721\pi\)
\(29\)	−2.59226	−	3.56794i	−0.481371	−	0.662550i	0.497397	−	0.867523i	\(-0.334289\pi\)
							−0.978768	+	0.204973i	\(0.934289\pi\)
\(31\)	−3.04947	−	2.21557i	−0.547702	−	0.397929i	0.279236	−	0.960223i	\(-0.409919\pi\)
							−0.826937	+	0.562294i	\(0.809919\pi\)
\(37\)	−1.24835	+	3.84202i	−0.205227	+	0.631625i	0.794477	+	0.607295i	\(0.207745\pi\)
							−0.999704	+	0.0243301i	\(0.992255\pi\)
\(41\)	−1.97884	+	6.09023i	−0.309042	+	0.951135i	0.669095	+	0.743177i	\(0.266682\pi\)
							−0.978138	+	0.207958i	\(0.933318\pi\)
\(43\)	10.9079			1.66343			0.831717	−	0.555199i	\(-0.187358\pi\)
							0.831717	+	0.555199i	\(0.187358\pi\)
\(47\)	1.67150	+	2.30062i	0.243813	+	0.335580i	0.913333	−	0.407214i	\(-0.133500\pi\)
							−0.669519	+	0.742795i	\(0.733500\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.2.o.a.69.7	yes	112
4.3	odd	2		800.2.be.a.369.1		112
5.2	odd	4		1000.2.t.b.901.40		224
5.3	odd	4		1000.2.t.b.901.17		224
5.4	even	2		1000.2.o.a.349.22		112
8.3	odd	2		800.2.be.a.369.28		112
8.5	even	2	inner	200.2.o.a.69.13	yes	112
25.3	odd	20		1000.2.t.b.101.51		224
25.4	even	10	inner	200.2.o.a.29.13	yes	112
25.21	even	5		1000.2.o.a.149.16		112
25.22	odd	20		1000.2.t.b.101.6		224
40.13	odd	4		1000.2.t.b.901.51		224
40.29	even	2		1000.2.o.a.349.16		112
40.37	odd	4		1000.2.t.b.901.6		224
100.79	odd	10		800.2.be.a.529.28		112
200.21	even	10		1000.2.o.a.149.22		112
200.29	even	10	inner	200.2.o.a.29.7	✓	112
200.53	odd	20		1000.2.t.b.101.17		224
200.179	odd	10		800.2.be.a.529.1		112
200.197	odd	20		1000.2.t.b.101.40		224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.2.o.a.29.7	✓	112	200.29	even	10	inner
200.2.o.a.29.13	yes	112	25.4	even	10	inner
200.2.o.a.69.7	yes	112	1.1	even	1	trivial
200.2.o.a.69.13	yes	112	8.5	even	2	inner
800.2.be.a.369.1		112	4.3	odd	2
800.2.be.a.369.28		112	8.3	odd	2
800.2.be.a.529.1		112	200.179	odd	10
800.2.be.a.529.28		112	100.79	odd	10
1000.2.o.a.149.16		112	25.21	even	5
1000.2.o.a.149.22		112	200.21	even	10
1000.2.o.a.349.16		112	40.29	even	2
1000.2.o.a.349.22		112	5.4	even	2
1000.2.t.b.101.6		224	25.22	odd	20
1000.2.t.b.101.17		224	200.53	odd	20
1000.2.t.b.101.40		224	200.197	odd	20
1000.2.t.b.101.51		224	25.3	odd	20
1000.2.t.b.901.6		224	40.37	odd	4
1000.2.t.b.901.17		224	5.3	odd	4
1000.2.t.b.901.40		224	5.2	odd	4
1000.2.t.b.901.51		224	40.13	odd	4