Embedded newform 200.2.o.a.69.19

• Label: 200.2.o.a.69.19
• 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			69.19
Character	\(\chi\)	\(=\)	200.69
Dual form			200.2.o.a.29.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.658957 - 1.25131i) q^{2} +(-2.03813 + 1.48079i) q^{3} +(-1.13155 - 1.64912i) q^{4} +(2.19360 - 0.433703i) q^{5} +(0.509884 + 3.52610i) q^{6} -3.58786i q^{7} +(-2.80920 + 0.329223i) q^{8} +(1.03418 - 3.18289i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 5 q^{2} - 3 q^{4} + q^{6} + 10 q^{8} - 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.658957	−	1.25131i	0.465953	−	0.884810i
							\(3\)	−2.03813	+	1.48079i	−1.17671	+	0.854932i	−0.991797	−	0.127822i	\(-0.959201\pi\)
−0.184916	+	0.982754i	\(0.559201\pi\)
\(5\)	2.19360	−	0.433703i	0.981010	−	0.193958i
							\(7\)		−	3.58786i		−	1.35608i	−0.735024	−	0.678042i	\(-0.762829\pi\)
0.735024	−	0.678042i	\(-0.237171\pi\)
\(11\)	2.49567	−	0.810891i	0.752472	−	0.244493i	0.0924272	−	0.995719i	\(-0.470537\pi\)
							0.660044	+	0.751227i	\(0.270537\pi\)
\(13\)	2.11444	−	6.50757i	0.586440	−	1.80488i	−0.00697033	−	0.999976i	\(-0.502219\pi\)
							0.593410	−	0.804900i	\(-0.297781\pi\)
\(17\)	−1.14335	+	1.57369i	−0.277304	+	0.381677i	−0.924839	−	0.380360i	\(-0.875800\pi\)
							0.647534	+	0.762036i	\(0.275800\pi\)
\(19\)	−3.00859	+	4.14097i	−0.690218	+	0.950003i	−1.00000		0.000804663i	\(-0.999744\pi\)
							0.309782	+	0.950808i	\(0.399744\pi\)
\(23\)	−0.737902	+	0.239759i	−0.153863	+	0.0499932i	−0.384936	−	0.922943i	\(-0.625776\pi\)
							0.231073	+	0.972936i	\(0.425776\pi\)
\(29\)	3.28631	+	4.52322i	0.610252	+	0.839940i	0.996598	−	0.0824137i	\(-0.0262629\pi\)
							−0.386346	+	0.922354i	\(0.626263\pi\)
\(31\)	1.72253	+	1.25149i	0.309376	+	0.224775i	0.731629	−	0.681704i	\(-0.238761\pi\)
							−0.422253	+	0.906478i	\(0.638761\pi\)
\(37\)	−2.73986	+	8.43241i	−0.450430	+	1.38628i	0.425988	+	0.904729i	\(0.359926\pi\)
							−0.876418	+	0.481551i	\(0.840074\pi\)
\(41\)	1.12308	−	3.45647i	0.175395	−	0.539811i	−0.824256	−	0.566217i	\(-0.808406\pi\)
							0.999651	+	0.0264065i	\(0.00840642\pi\)
\(43\)	0.652170			0.0994550			0.0497275	−	0.998763i	\(-0.484165\pi\)
							0.0497275	+	0.998763i	\(0.484165\pi\)
\(47\)	0.541317	+	0.745059i	0.0789592	+	0.108678i	0.846670	−	0.532119i	\(-0.178604\pi\)
							−0.767710	+	0.640797i	\(0.778604\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.2.o.a.69.19	yes	112
4.3	odd	2		800.2.be.a.369.25		112
5.2	odd	4		1000.2.t.b.901.48		224
5.3	odd	4		1000.2.t.b.901.9		224
5.4	even	2		1000.2.o.a.349.10		112
8.3	odd	2		800.2.be.a.369.4		112
8.5	even	2	inner	200.2.o.a.69.24	yes	112
25.3	odd	20		1000.2.t.b.101.37		224
25.4	even	10	inner	200.2.o.a.29.24	yes	112
25.21	even	5		1000.2.o.a.149.5		112
25.22	odd	20		1000.2.t.b.101.20		224
40.13	odd	4		1000.2.t.b.901.37		224
40.29	even	2		1000.2.o.a.349.5		112
40.37	odd	4		1000.2.t.b.901.20		224
100.79	odd	10		800.2.be.a.529.4		112
200.21	even	10		1000.2.o.a.149.10		112
200.29	even	10	inner	200.2.o.a.29.19	✓	112
200.53	odd	20		1000.2.t.b.101.9		224
200.179	odd	10		800.2.be.a.529.25		112
200.197	odd	20		1000.2.t.b.101.48		224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.2.o.a.29.19	✓	112	200.29	even	10	inner
200.2.o.a.29.24	yes	112	25.4	even	10	inner
200.2.o.a.69.19	yes	112	1.1	even	1	trivial
200.2.o.a.69.24	yes	112	8.5	even	2	inner
800.2.be.a.369.4		112	8.3	odd	2
800.2.be.a.369.25		112	4.3	odd	2
800.2.be.a.529.4		112	100.79	odd	10
800.2.be.a.529.25		112	200.179	odd	10
1000.2.o.a.149.5		112	25.21	even	5
1000.2.o.a.149.10		112	200.21	even	10
1000.2.o.a.349.5		112	40.29	even	2
1000.2.o.a.349.10		112	5.4	even	2
1000.2.t.b.101.9		224	200.53	odd	20
1000.2.t.b.101.20		224	25.22	odd	20
1000.2.t.b.101.37		224	25.3	odd	20
1000.2.t.b.101.48		224	200.197	odd	20
1000.2.t.b.901.9		224	5.3	odd	4
1000.2.t.b.901.20		224	40.37	odd	4
1000.2.t.b.901.37		224	40.13	odd	4
1000.2.t.b.901.48		224	5.2	odd	4