Embedded newform 200.4.a.g.1.1

• Label: 200.4.a.g.1.1
• Level: $200$
• Weight: $4$
• Character: 200.1
• Self dual: yes
• Analytic conductor: $11.800$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	200.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(11.8003820011\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	200.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+4.00000 q^{3} -24.0000 q^{7} -11.0000 q^{9} -44.0000 q^{11} -22.0000 q^{13} -50.0000 q^{17} +44.0000 q^{19} -96.0000 q^{21} +56.0000 q^{23} -152.000 q^{27} +198.000 q^{29} -160.000 q^{31} -176.000 q^{33} +162.000 q^{37} -88.0000 q^{39} -198.000 q^{41} -52.0000 q^{43} -528.000 q^{47} +233.000 q^{49} -200.000 q^{51} +242.000 q^{53} +176.000 q^{57} -668.000 q^{59} +550.000 q^{61} +264.000 q^{63} -188.000 q^{67} +224.000 q^{69} +728.000 q^{71} -154.000 q^{73} +1056.00 q^{77} -656.000 q^{79} -311.000 q^{81} -236.000 q^{83} +792.000 q^{87} +714.000 q^{89} +528.000 q^{91} -640.000 q^{93} +478.000 q^{97} +484.000 q^{99} +O(q^{100})\)

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	0
			\(3\)	4.00000	0.769800	0.384900	−	0.922958i	\(-0.374236\pi\)
0.384900	+	0.922958i				\(0.374236\pi\)
\(5\)	0	0
			\(7\)	−24.0000	−1.29588	−0.647939	−	0.761692i	\(-0.724369\pi\)
−0.647939	+	0.761692i				\(0.724369\pi\)
\(11\)	−44.0000	−1.20605	−0.603023	−	0.797724i	\(-0.706037\pi\)
			−0.603023	+	0.797724i	\(0.706037\pi\)
\(13\)	−22.0000	−0.469362	−0.234681	−	0.972072i	\(-0.575405\pi\)
			−0.234681	+	0.972072i	\(0.575405\pi\)
\(17\)	−50.0000	−0.713340	−0.356670	−	0.934230i	\(-0.616088\pi\)
			−0.356670	+	0.934230i	\(0.616088\pi\)
\(19\)	44.0000	0.531279	0.265639	−	0.964072i	\(-0.414417\pi\)
			0.265639	+	0.964072i	\(0.414417\pi\)
\(23\)	56.0000	0.507687	0.253844	−	0.967245i	\(-0.418305\pi\)
			0.253844	+	0.967245i	\(0.418305\pi\)
\(29\)	198.000	1.26785	0.633925	−	0.773394i	\(-0.281443\pi\)
			0.633925	+	0.773394i	\(0.281443\pi\)
\(31\)	−160.000	−0.926995	−0.463498	−	0.886098i	\(-0.653406\pi\)
			−0.463498	+	0.886098i	\(0.653406\pi\)
\(37\)	162.000	0.719801	0.359900	−	0.932991i	\(-0.382811\pi\)
			0.359900	+	0.932991i	\(0.382811\pi\)
\(41\)	−198.000	−0.754205	−0.377102	−	0.926172i	\(-0.623080\pi\)
			−0.377102	+	0.926172i	\(0.623080\pi\)
\(43\)	−52.0000	−0.184417	−0.0922084	−	0.995740i	\(-0.529393\pi\)
			−0.0922084	+	0.995740i	\(0.529393\pi\)
\(47\)	−528.000	−1.63865	−0.819327	−	0.573327i	\(-0.805653\pi\)
			−0.819327	+	0.573327i	\(0.805653\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.4.a.g.1.1		1
3.2	odd	2		1800.4.a.d.1.1		1
4.3	odd	2		400.4.a.g.1.1		1
5.2	odd	4		200.4.c.e.49.1		2
5.3	odd	4		200.4.c.e.49.2		2
5.4	even	2		8.4.a.a.1.1	✓	1
8.3	odd	2		1600.4.a.bm.1.1		1
8.5	even	2		1600.4.a.o.1.1		1
15.2	even	4		1800.4.f.u.649.1		2
15.8	even	4		1800.4.f.u.649.2		2
15.14	odd	2		72.4.a.c.1.1		1
20.3	even	4		400.4.c.i.49.1		2
20.7	even	4		400.4.c.i.49.2		2
20.19	odd	2		16.4.a.a.1.1		1
35.4	even	6		392.4.i.g.177.1		2
35.9	even	6		392.4.i.g.361.1		2
35.19	odd	6		392.4.i.b.361.1		2
35.24	odd	6		392.4.i.b.177.1		2
35.34	odd	2		392.4.a.e.1.1		1
40.19	odd	2		64.4.a.b.1.1		1
40.29	even	2		64.4.a.d.1.1		1
45.4	even	6		648.4.i.h.217.1		2
45.14	odd	6		648.4.i.e.217.1		2
45.29	odd	6		648.4.i.e.433.1		2
45.34	even	6		648.4.i.h.433.1		2
55.54	odd	2		968.4.a.a.1.1		1
60.59	even	2		144.4.a.e.1.1		1
65.64	even	2		1352.4.a.a.1.1		1
80.19	odd	4		256.4.b.g.129.2		2
80.29	even	4		256.4.b.a.129.1		2
80.59	odd	4		256.4.b.g.129.1		2
80.69	even	4		256.4.b.a.129.2		2
85.84	even	2		2312.4.a.a.1.1		1
120.29	odd	2		576.4.a.k.1.1		1
120.59	even	2		576.4.a.j.1.1		1
140.139	even	2		784.4.a.e.1.1		1
220.219	even	2		1936.4.a.l.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.4.a.a.1.1	✓	1	5.4	even	2
16.4.a.a.1.1		1	20.19	odd	2
64.4.a.b.1.1		1	40.19	odd	2
64.4.a.d.1.1		1	40.29	even	2
72.4.a.c.1.1		1	15.14	odd	2
144.4.a.e.1.1		1	60.59	even	2
200.4.a.g.1.1		1	1.1	even	1	trivial
200.4.c.e.49.1		2	5.2	odd	4
200.4.c.e.49.2		2	5.3	odd	4
256.4.b.a.129.1		2	80.29	even	4
256.4.b.a.129.2		2	80.69	even	4
256.4.b.g.129.1		2	80.59	odd	4
256.4.b.g.129.2		2	80.19	odd	4
392.4.a.e.1.1		1	35.34	odd	2
392.4.i.b.177.1		2	35.24	odd	6
392.4.i.b.361.1		2	35.19	odd	6
392.4.i.g.177.1		2	35.4	even	6
392.4.i.g.361.1		2	35.9	even	6
400.4.a.g.1.1		1	4.3	odd	2
400.4.c.i.49.1		2	20.3	even	4
400.4.c.i.49.2		2	20.7	even	4
576.4.a.j.1.1		1	120.59	even	2
576.4.a.k.1.1		1	120.29	odd	2
648.4.i.e.217.1		2	45.14	odd	6
648.4.i.e.433.1		2	45.29	odd	6
648.4.i.h.217.1		2	45.4	even	6
648.4.i.h.433.1		2	45.34	even	6
784.4.a.e.1.1		1	140.139	even	2
968.4.a.a.1.1		1	55.54	odd	2
1352.4.a.a.1.1		1	65.64	even	2
1600.4.a.o.1.1		1	8.5	even	2
1600.4.a.bm.1.1		1	8.3	odd	2
1800.4.a.d.1.1		1	3.2	odd	2
1800.4.f.u.649.1		2	15.2	even	4
1800.4.f.u.649.2		2	15.8	even	4
1936.4.a.l.1.1		1	220.219	even	2
2312.4.a.a.1.1		1	85.84	even	2