Embedded newform 3200.2.a.u.1.1

• Label: 3200.2.a.u.1.1
• Level: $3200$
• Weight: $2$
• Character: 3200.1
• Self dual: yes
• Analytic conductor: $25.552$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(25.5521286468\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 128)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	3200.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+2.00000 q^{3} -4.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} -2.00000 q^{13} +2.00000 q^{17} -2.00000 q^{19} -8.00000 q^{21} +4.00000 q^{23} -4.00000 q^{27} -6.00000 q^{29} +4.00000 q^{33} -10.0000 q^{37} -4.00000 q^{39} -6.00000 q^{41} +6.00000 q^{43} -8.00000 q^{47} +9.00000 q^{49} +4.00000 q^{51} +6.00000 q^{53} -4.00000 q^{57} -14.0000 q^{59} +2.00000 q^{61} -4.00000 q^{63} +10.0000 q^{67} +8.00000 q^{69} -12.0000 q^{71} -14.0000 q^{73} -8.00000 q^{77} +8.00000 q^{79} -11.0000 q^{81} -6.00000 q^{83} -12.0000 q^{87} -2.00000 q^{89} +8.00000 q^{91} +2.00000 q^{97} +2.00000 q^{99} +O(q^{100})\)

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
			\(3\)	2.00000	1.15470	0.577350	−	0.816497i	\(-0.304087\pi\)
0.577350	+	0.816497i				\(0.304087\pi\)
\(5\)	0	0
			\(7\)	−4.00000	−1.51186	−0.755929	−	0.654654i	\(-0.772814\pi\)
−0.755929	+	0.654654i				\(0.772814\pi\)
\(11\)	2.00000	0.603023	0.301511	−	0.953463i	\(-0.402509\pi\)
			0.301511	+	0.953463i	\(0.402509\pi\)
\(13\)	−2.00000	−0.554700	−0.277350	−	0.960769i	\(-0.589456\pi\)
			−0.277350	+	0.960769i	\(0.589456\pi\)
\(17\)	2.00000	0.485071	0.242536	−	0.970143i	\(-0.422021\pi\)
			0.242536	+	0.970143i	\(0.422021\pi\)
\(19\)	−2.00000	−0.458831	−0.229416	−	0.973329i	\(-0.573682\pi\)
			−0.229416	+	0.973329i	\(0.573682\pi\)
\(23\)	4.00000	0.834058	0.417029	−	0.908893i	\(-0.363071\pi\)
			0.417029	+	0.908893i	\(0.363071\pi\)
\(29\)	−6.00000	−1.11417	−0.557086	−	0.830455i	\(-0.688081\pi\)
			−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(37\)	−10.0000	−1.64399	−0.821995	−	0.569495i	\(-0.807139\pi\)
			−0.821995	+	0.569495i	\(0.807139\pi\)
\(41\)	−6.00000	−0.937043	−0.468521	−	0.883452i	\(-0.655213\pi\)
			−0.468521	+	0.883452i	\(0.655213\pi\)
\(43\)	6.00000	0.914991	0.457496	−	0.889212i	\(-0.348747\pi\)
			0.457496	+	0.889212i	\(0.348747\pi\)
\(47\)	−8.00000	−1.16692	−0.583460	−	0.812142i	\(-0.698301\pi\)
			−0.583460	+	0.812142i	\(0.698301\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3200.2.a.u.1.1		1
4.3	odd	2		3200.2.a.h.1.1		1
5.2	odd	4		3200.2.c.k.2049.1		2
5.3	odd	4		3200.2.c.k.2049.2		2
5.4	even	2		128.2.a.b.1.1	yes	1
8.3	odd	2		3200.2.a.x.1.1		1
8.5	even	2		3200.2.a.e.1.1		1
15.14	odd	2		1152.2.a.h.1.1		1
20.3	even	4		3200.2.c.e.2049.1		2
20.7	even	4		3200.2.c.e.2049.2		2
20.19	odd	2		128.2.a.d.1.1	yes	1
35.34	odd	2		6272.2.a.g.1.1		1
40.3	even	4		3200.2.c.l.2049.2		2
40.13	odd	4		3200.2.c.f.2049.1		2
40.19	odd	2		128.2.a.a.1.1	✓	1
40.27	even	4		3200.2.c.l.2049.1		2
40.29	even	2		128.2.a.c.1.1	yes	1
40.37	odd	4		3200.2.c.f.2049.2		2
60.59	even	2		1152.2.a.c.1.1		1
80.19	odd	4		256.2.b.c.129.2		2
80.29	even	4		256.2.b.a.129.1		2
80.59	odd	4		256.2.b.c.129.1		2
80.69	even	4		256.2.b.a.129.2		2
120.29	odd	2		1152.2.a.r.1.1		1
120.59	even	2		1152.2.a.m.1.1		1
140.139	even	2		6272.2.a.a.1.1		1
160.19	odd	8		1024.2.e.i.257.1		4
160.29	even	8		1024.2.e.m.257.1		4
160.59	odd	8		1024.2.e.i.769.1		4
160.69	even	8		1024.2.e.m.769.2		4
160.99	odd	8		1024.2.e.i.257.2		4
160.109	even	8		1024.2.e.m.257.2		4
160.139	odd	8		1024.2.e.i.769.2		4
160.149	even	8		1024.2.e.m.769.1		4
240.29	odd	4		2304.2.d.b.1153.2		2
240.59	even	4		2304.2.d.r.1153.1		2
240.149	odd	4		2304.2.d.b.1153.1		2
240.179	even	4		2304.2.d.r.1153.2		2
280.69	odd	2		6272.2.a.b.1.1		1
280.139	even	2		6272.2.a.h.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.2.a.a.1.1	✓	1	40.19	odd	2
128.2.a.b.1.1	yes	1	5.4	even	2
128.2.a.c.1.1	yes	1	40.29	even	2
128.2.a.d.1.1	yes	1	20.19	odd	2
256.2.b.a.129.1		2	80.29	even	4
256.2.b.a.129.2		2	80.69	even	4
256.2.b.c.129.1		2	80.59	odd	4
256.2.b.c.129.2		2	80.19	odd	4
1024.2.e.i.257.1		4	160.19	odd	8
1024.2.e.i.257.2		4	160.99	odd	8
1024.2.e.i.769.1		4	160.59	odd	8
1024.2.e.i.769.2		4	160.139	odd	8
1024.2.e.m.257.1		4	160.29	even	8
1024.2.e.m.257.2		4	160.109	even	8
1024.2.e.m.769.1		4	160.149	even	8
1024.2.e.m.769.2		4	160.69	even	8
1152.2.a.c.1.1		1	60.59	even	2
1152.2.a.h.1.1		1	15.14	odd	2
1152.2.a.m.1.1		1	120.59	even	2
1152.2.a.r.1.1		1	120.29	odd	2
2304.2.d.b.1153.1		2	240.149	odd	4
2304.2.d.b.1153.2		2	240.29	odd	4
2304.2.d.r.1153.1		2	240.59	even	4
2304.2.d.r.1153.2		2	240.179	even	4
3200.2.a.e.1.1		1	8.5	even	2
3200.2.a.h.1.1		1	4.3	odd	2
3200.2.a.u.1.1		1	1.1	even	1	trivial
3200.2.a.x.1.1		1	8.3	odd	2
3200.2.c.e.2049.1		2	20.3	even	4
3200.2.c.e.2049.2		2	20.7	even	4
3200.2.c.f.2049.1		2	40.13	odd	4
3200.2.c.f.2049.2		2	40.37	odd	4
3200.2.c.k.2049.1		2	5.2	odd	4
3200.2.c.k.2049.2		2	5.3	odd	4
3200.2.c.l.2049.1		2	40.27	even	4
3200.2.c.l.2049.2		2	40.3	even	4
6272.2.a.a.1.1		1	140.139	even	2
6272.2.a.b.1.1		1	280.69	odd	2
6272.2.a.g.1.1		1	35.34	odd	2
6272.2.a.h.1.1		1	280.139	even	2