Embedded newform 200.2.o.a.69.9

• Label: 200.2.o.a.69.9
• Level: $200$
• Weight: $2$
• Character: 200.69
• Analytic conductor: $1.597$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• 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			69.9
Character	\(\chi\)	\(=\)	200.69
Dual form			200.2.o.a.29.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.908199 + 1.08405i) q^{2} +(2.01785 - 1.46605i) q^{3} +(-0.350349 - 1.96907i) q^{4} +(-1.86593 - 1.23219i) q^{5} +(-0.243326 + 3.51892i) q^{6} -0.110917i q^{7} +(2.45277 + 1.40851i) q^{8} +(0.995348 - 3.06337i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 5 q^{2} - 3 q^{4} + q^{6} + 10 q^{8} - 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.908199	+	1.08405i	−0.642194	+	0.766542i
							\(3\)	2.01785	−	1.46605i	1.16500	−	0.846425i	0.174602	−	0.984639i	\(-0.444136\pi\)
0.990402	+	0.138214i	\(0.0441361\pi\)
\(5\)	−1.86593	−	1.23219i	−0.834470	−	0.551054i
							\(7\)		−	0.110917i		−	0.0419227i	−0.999780	−	0.0209613i	\(-0.993327\pi\)
0.999780	−	0.0209613i	\(-0.00667269\pi\)
\(11\)	1.63700	−	0.531893i	0.493574	−	0.160372i	−0.0516452	−	0.998665i	\(-0.516446\pi\)
							0.545219	+	0.838294i	\(0.316446\pi\)
\(13\)	1.48445	−	4.56868i	0.411713	−	1.26712i	−0.503445	−	0.864027i	\(-0.667934\pi\)
							0.915158	−	0.403096i	\(-0.132066\pi\)
\(17\)	0.269588	−	0.371056i	0.0653848	−	0.0899944i	−0.775072	−	0.631873i	\(-0.782287\pi\)
							0.840457	+	0.541878i	\(0.182287\pi\)
\(19\)	2.40052	−	3.30403i	0.550716	−	0.757996i	−0.439393	−	0.898295i	\(-0.644806\pi\)
							0.990109	+	0.140299i	\(0.0448064\pi\)
\(23\)	−6.10792	+	1.98459i	−1.27359	+	0.413815i	−0.866318	−	0.499492i	\(-0.833520\pi\)
							−0.407272	+	0.913307i	\(0.633520\pi\)
\(29\)	5.29624	+	7.28964i	0.983486	+	1.35365i	0.934930	+	0.354833i	\(0.115462\pi\)
							0.0485566	+	0.998820i	\(0.484538\pi\)
\(31\)	2.87291	+	2.08729i	0.515990	+	0.374889i	0.815091	−	0.579333i	\(-0.196687\pi\)
							−0.299101	+	0.954221i	\(0.596687\pi\)
\(37\)	−0.500287	+	1.53972i	−0.0822467	+	0.253129i	−0.983721	−	0.179704i	\(-0.942486\pi\)
							0.901474	+	0.432833i	\(0.142486\pi\)
\(41\)	−3.51556	+	10.8198i	−0.549039	+	1.68977i	0.162149	+	0.986766i	\(0.448157\pi\)
							−0.711188	+	0.703002i	\(0.751843\pi\)
\(43\)	−10.5304			−1.60588			−0.802938	−	0.596063i	\(-0.796731\pi\)
							−0.802938	+	0.596063i	\(0.796731\pi\)
\(47\)	1.82373	+	2.51014i	0.266018	+	0.366142i	0.921040	−	0.389467i	\(-0.127341\pi\)
							−0.655022	+	0.755609i	\(0.727341\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.2.o.a.69.9	yes	112
4.3	odd	2		800.2.be.a.369.5		112
5.2	odd	4		1000.2.t.b.901.14		224
5.3	odd	4		1000.2.t.b.901.43		224
5.4	even	2		1000.2.o.a.349.20		112
8.3	odd	2		800.2.be.a.369.24		112
8.5	even	2	inner	200.2.o.a.69.2	yes	112
25.3	odd	20		1000.2.t.b.101.27		224
25.4	even	10	inner	200.2.o.a.29.2	✓	112
25.21	even	5		1000.2.o.a.149.27		112
25.22	odd	20		1000.2.t.b.101.30		224
40.13	odd	4		1000.2.t.b.901.27		224
40.29	even	2		1000.2.o.a.349.27		112
40.37	odd	4		1000.2.t.b.901.30		224
100.79	odd	10		800.2.be.a.529.24		112
200.21	even	10		1000.2.o.a.149.20		112
200.29	even	10	inner	200.2.o.a.29.9	yes	112
200.53	odd	20		1000.2.t.b.101.43		224
200.179	odd	10		800.2.be.a.529.5		112
200.197	odd	20		1000.2.t.b.101.14		224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.2.o.a.29.2	✓	112	25.4	even	10	inner
200.2.o.a.29.9	yes	112	200.29	even	10	inner
200.2.o.a.69.2	yes	112	8.5	even	2	inner
200.2.o.a.69.9	yes	112	1.1	even	1	trivial
800.2.be.a.369.5		112	4.3	odd	2
800.2.be.a.369.24		112	8.3	odd	2
800.2.be.a.529.5		112	200.179	odd	10
800.2.be.a.529.24		112	100.79	odd	10
1000.2.o.a.149.20		112	200.21	even	10
1000.2.o.a.149.27		112	25.21	even	5
1000.2.o.a.349.20		112	5.4	even	2
1000.2.o.a.349.27		112	40.29	even	2
1000.2.t.b.101.14		224	200.197	odd	20
1000.2.t.b.101.27		224	25.3	odd	20
1000.2.t.b.101.30		224	25.22	odd	20
1000.2.t.b.101.43		224	200.53	odd	20
1000.2.t.b.901.14		224	5.2	odd	4
1000.2.t.b.901.27		224	40.13	odd	4
1000.2.t.b.901.30		224	40.37	odd	4
1000.2.t.b.901.43		224	5.3	odd	4