Embedded newform 200.2.o.a.69.8

• Label: 200.2.o.a.69.8
• 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			69.8
Character	\(\chi\)	\(=\)	200.69
Dual form			200.2.o.a.29.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.01240 - 0.987445i) q^{2} +(1.39188 - 1.01126i) q^{3} +(0.0499066 + 1.99938i) q^{4} +(0.122399 + 2.23272i) q^{5} +(-2.40770 - 0.350605i) q^{6} -4.42380i q^{7} +(1.92375 - 2.07345i) q^{8} +(-0.0123697 + 0.0380700i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 5 q^{2} - 3 q^{4} + q^{6} + 10 q^{8} - 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.01240	−	0.987445i	−0.715875	−	0.698229i
							\(3\)	1.39188	−	1.01126i	0.803601	−	0.583851i	−0.108367	−	0.994111i	\(-0.534562\pi\)
0.911968	+	0.410260i	\(0.134562\pi\)
\(5\)	0.122399	+	2.23272i	0.0547384	+	0.998501i
							\(7\)		−	4.42380i		−	1.67204i	−0.548699	−	0.836020i	\(-0.684877\pi\)
0.548699	−	0.836020i	\(-0.315123\pi\)
\(11\)	5.07953	−	1.65044i	1.53153	−	0.497626i	0.582509	−	0.812824i	\(-0.302071\pi\)
							0.949026	+	0.315199i	\(0.102071\pi\)
\(13\)	−0.237511	+	0.730985i	−0.0658738	+	0.202739i	−0.978576	−	0.205887i	\(-0.933992\pi\)
							0.912702	+	0.408626i	\(0.133992\pi\)
\(17\)	3.34610	−	4.60551i	0.811549	−	1.11700i	−0.179534	−	0.983752i	\(-0.557459\pi\)
							0.991083	−	0.133249i	\(-0.0425410\pi\)
\(19\)	−1.01760	+	1.40061i	−0.233454	+	0.321322i	−0.909631	−	0.415418i	\(-0.863635\pi\)
							0.676177	+	0.736739i	\(0.263635\pi\)
\(23\)	−5.01464	+	1.62935i	−1.04562	+	0.339744i	−0.780950	−	0.624594i	\(-0.785265\pi\)
							−0.264675	+	0.964338i	\(0.585265\pi\)
\(29\)	1.21119	+	1.66707i	0.224913	+	0.309567i	0.906529	−	0.422144i	\(-0.138722\pi\)
							−0.681616	+	0.731710i	\(0.738722\pi\)
\(31\)	−2.10890	−	1.53221i	−0.378770	−	0.275192i	0.382068	−	0.924134i	\(-0.375212\pi\)
							−0.760838	+	0.648942i	\(0.775212\pi\)
\(37\)	−2.01004	+	6.18628i	−0.330449	+	1.01702i	0.638471	+	0.769646i	\(0.279567\pi\)
							−0.968921	+	0.247372i	\(0.920433\pi\)
\(41\)	−0.754681	+	2.32267i	−0.117861	+	0.362740i	−0.992533	−	0.121976i	\(-0.961077\pi\)
							0.874672	+	0.484716i	\(0.161077\pi\)
\(43\)	−6.65200			−1.01442			−0.507210	−	0.861822i	\(-0.669323\pi\)
							−0.507210	+	0.861822i	\(0.669323\pi\)
\(47\)	4.72720	+	6.50644i	0.689534	+	0.949062i	0.999999	−	0.00152305i	\(-0.000484802\pi\)
							−0.310465	+	0.950585i	\(0.600485\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.2.o.a.69.8	yes	112
4.3	odd	2		800.2.be.a.369.7		112
5.2	odd	4		1000.2.t.b.901.41		224
5.3	odd	4		1000.2.t.b.901.16		224
5.4	even	2		1000.2.o.a.349.21		112
8.3	odd	2		800.2.be.a.369.22		112
8.5	even	2	inner	200.2.o.a.69.14	yes	112
25.3	odd	20		1000.2.t.b.101.53		224
25.4	even	10	inner	200.2.o.a.29.14	yes	112
25.21	even	5		1000.2.o.a.149.15		112
25.22	odd	20		1000.2.t.b.101.4		224
40.13	odd	4		1000.2.t.b.901.53		224
40.29	even	2		1000.2.o.a.349.15		112
40.37	odd	4		1000.2.t.b.901.4		224
100.79	odd	10		800.2.be.a.529.22		112
200.21	even	10		1000.2.o.a.149.21		112
200.29	even	10	inner	200.2.o.a.29.8	✓	112
200.53	odd	20		1000.2.t.b.101.16		224
200.179	odd	10		800.2.be.a.529.7		112
200.197	odd	20		1000.2.t.b.101.41		224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.2.o.a.29.8	✓	112	200.29	even	10	inner
200.2.o.a.29.14	yes	112	25.4	even	10	inner
200.2.o.a.69.8	yes	112	1.1	even	1	trivial
200.2.o.a.69.14	yes	112	8.5	even	2	inner
800.2.be.a.369.7		112	4.3	odd	2
800.2.be.a.369.22		112	8.3	odd	2
800.2.be.a.529.7		112	200.179	odd	10
800.2.be.a.529.22		112	100.79	odd	10
1000.2.o.a.149.15		112	25.21	even	5
1000.2.o.a.149.21		112	200.21	even	10
1000.2.o.a.349.15		112	40.29	even	2
1000.2.o.a.349.21		112	5.4	even	2
1000.2.t.b.101.4		224	25.22	odd	20
1000.2.t.b.101.16		224	200.53	odd	20
1000.2.t.b.101.41		224	200.197	odd	20
1000.2.t.b.101.53		224	25.3	odd	20
1000.2.t.b.901.4		224	40.37	odd	4
1000.2.t.b.901.16		224	5.3	odd	4
1000.2.t.b.901.41		224	5.2	odd	4
1000.2.t.b.901.53		224	40.13	odd	4