Embedded newform 200.2.o.a.69.10

• Label: 200.2.o.a.69.10
• 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			69.10
Character	\(\chi\)	\(=\)	200.69
Dual form			200.2.o.a.29.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.717591 - 1.21863i) q^{2} +(-2.54275 + 1.84742i) q^{3} +(-0.970125 + 1.74896i) q^{4} +(-1.08744 - 1.95384i) q^{5} +(4.07598 + 1.77299i) q^{6} +1.17589i q^{7} +(2.82749 - 0.0728129i) q^{8} +(2.12559 - 6.54190i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 5 q^{2} - 3 q^{4} + q^{6} + 10 q^{8} - 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.717591	−	1.21863i	−0.507414	−	0.861703i
							\(3\)	−2.54275	+	1.84742i	−1.46806	+	1.06661i	−0.486890	+	0.873463i	\(0.661869\pi\)
−0.981170	+	0.193145i	\(0.938131\pi\)
\(5\)	−1.08744	−	1.95384i	−0.486318	−	0.873782i
							\(7\)			1.17589i			0.444443i	0.974996	+	0.222222i	\(0.0713308\pi\)
−0.974996	+	0.222222i	\(0.928669\pi\)
\(11\)	3.77756	−	1.22741i	1.13898	−	0.370077i	0.321997	−	0.946741i	\(-0.395646\pi\)
							0.816981	+	0.576664i	\(0.195646\pi\)
\(13\)	−0.280944	+	0.864656i	−0.0779198	+	0.239812i	−0.982427	−	0.186645i	\(-0.940239\pi\)
							0.904508	+	0.426457i	\(0.140239\pi\)
\(17\)	3.27511	−	4.50780i	0.794331	−	1.09330i	−0.199225	−	0.979954i	\(-0.563842\pi\)
							0.993555	−	0.113348i	\(-0.0361576\pi\)
\(19\)	2.64864	−	3.64555i	0.607641	−	0.836346i	−0.388740	−	0.921348i	\(-0.627090\pi\)
							0.996381	+	0.0850017i	\(0.0270896\pi\)
\(23\)	−2.07468	+	0.674105i	−0.432601	+	0.140561i	−0.517220	−	0.855853i	\(-0.673033\pi\)
							0.0846183	+	0.996413i	\(0.473033\pi\)
\(29\)	3.65841	+	5.03536i	0.679349	+	0.935044i	0.999926	−	0.0121788i	\(-0.00387674\pi\)
							−0.320577	+	0.947223i	\(0.603877\pi\)
\(31\)	−1.16778	−	0.848445i	−0.209740	−	0.152385i	0.477956	−	0.878384i	\(-0.341378\pi\)
							−0.687696	+	0.725999i	\(0.741378\pi\)
\(37\)	2.16303	−	6.65712i	0.355600	−	1.09442i	−0.600061	−	0.799954i	\(-0.704857\pi\)
							0.955661	−	0.294470i	\(-0.0951431\pi\)
\(41\)	1.97065	−	6.06503i	0.307763	−	0.947198i	−0.670868	−	0.741577i	\(-0.734078\pi\)
							0.978632	−	0.205622i	\(-0.0659216\pi\)
\(43\)	2.96089			0.451532			0.225766	−	0.974182i	\(-0.427512\pi\)
							0.225766	+	0.974182i	\(0.427512\pi\)
\(47\)	−3.12538	−	4.30172i	−0.455884	−	0.627470i	0.517765	−	0.855523i	\(-0.326764\pi\)
							−0.973649	+	0.228053i	\(0.926764\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.2.o.a.69.10	yes	112
4.3	odd	2		800.2.be.a.369.27		112
5.2	odd	4		1000.2.t.b.901.47		224
5.3	odd	4		1000.2.t.b.901.10		224
5.4	even	2		1000.2.o.a.349.19		112
8.3	odd	2		800.2.be.a.369.2		112
8.5	even	2	inner	200.2.o.a.69.16	yes	112
25.3	odd	20		1000.2.t.b.101.55		224
25.4	even	10	inner	200.2.o.a.29.16	yes	112
25.21	even	5		1000.2.o.a.149.13		112
25.22	odd	20		1000.2.t.b.101.2		224
40.13	odd	4		1000.2.t.b.901.55		224
40.29	even	2		1000.2.o.a.349.13		112
40.37	odd	4		1000.2.t.b.901.2		224
100.79	odd	10		800.2.be.a.529.2		112
200.21	even	10		1000.2.o.a.149.19		112
200.29	even	10	inner	200.2.o.a.29.10	✓	112
200.53	odd	20		1000.2.t.b.101.10		224
200.179	odd	10		800.2.be.a.529.27		112
200.197	odd	20		1000.2.t.b.101.47		224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.2.o.a.29.10	✓	112	200.29	even	10	inner
200.2.o.a.29.16	yes	112	25.4	even	10	inner
200.2.o.a.69.10	yes	112	1.1	even	1	trivial
200.2.o.a.69.16	yes	112	8.5	even	2	inner
800.2.be.a.369.2		112	8.3	odd	2
800.2.be.a.369.27		112	4.3	odd	2
800.2.be.a.529.2		112	100.79	odd	10
800.2.be.a.529.27		112	200.179	odd	10
1000.2.o.a.149.13		112	25.21	even	5
1000.2.o.a.149.19		112	200.21	even	10
1000.2.o.a.349.13		112	40.29	even	2
1000.2.o.a.349.19		112	5.4	even	2
1000.2.t.b.101.2		224	25.22	odd	20
1000.2.t.b.101.10		224	200.53	odd	20
1000.2.t.b.101.47		224	200.197	odd	20
1000.2.t.b.101.55		224	25.3	odd	20
1000.2.t.b.901.2		224	40.37	odd	4
1000.2.t.b.901.10		224	5.3	odd	4
1000.2.t.b.901.47		224	5.2	odd	4
1000.2.t.b.901.55		224	40.13	odd	4