Embedded newform 1152.2.a.m.1.1

• Label: 1152.2.a.m.1.1
• Level: $1152$
• Weight: $2$
• Character: 1152.1
• Self dual: yes
• Analytic conductor: $9.199$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1152.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(9.19876631285\)
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	\(\chi\)	\(=\)	1152.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+2.00000 q^{5} -4.00000 q^{7} -2.00000 q^{11} -2.00000 q^{13} +2.00000 q^{17} -2.00000 q^{19} -4.00000 q^{23} -1.00000 q^{25} -6.00000 q^{29} -8.00000 q^{35} -10.0000 q^{37} +6.00000 q^{41} -6.00000 q^{43} +8.00000 q^{47} +9.00000 q^{49} -6.00000 q^{53} -4.00000 q^{55} +14.0000 q^{59} -2.00000 q^{61} -4.00000 q^{65} -10.0000 q^{67} -12.0000 q^{71} +14.0000 q^{73} +8.00000 q^{77} -8.00000 q^{79} -6.00000 q^{83} +4.00000 q^{85} +2.00000 q^{89} +8.00000 q^{91} -4.00000 q^{95} -2.00000 q^{97} +O(q^{100})\)

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
			\(3\)	0	0
\(5\)	2.00000	0.894427				0.447214	−	0.894427i	\(-0.352416\pi\)
			0.447214	+	0.894427i	\(0.352416\pi\)
\(7\)	−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	\(-0.597491\pi\)
			−0.301511	+	0.953463i	\(0.597491\pi\)
\(13\)	−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	\(-0.636929\pi\)
			−0.417029	+	0.908893i	\(0.636929\pi\)
\(29\)	−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	\(-0.344787\pi\)
			0.468521	+	0.883452i	\(0.344787\pi\)
\(43\)	−6.00000	−0.914991	−0.457496	−	0.889212i	\(-0.651253\pi\)
			−0.457496	+	0.889212i	\(0.651253\pi\)
\(47\)	8.00000	1.16692	0.583460	−	0.812142i	\(-0.301699\pi\)
			0.583460	+	0.812142i	\(0.301699\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.2.a.m.1.1		1
3.2	odd	2		128.2.a.a.1.1	✓	1
4.3	odd	2		1152.2.a.r.1.1		1
8.3	odd	2		1152.2.a.h.1.1		1
8.5	even	2		1152.2.a.c.1.1		1
12.11	even	2		128.2.a.c.1.1	yes	1
15.2	even	4		3200.2.c.l.2049.2		2
15.8	even	4		3200.2.c.l.2049.1		2
15.14	odd	2		3200.2.a.x.1.1		1
16.3	odd	4		2304.2.d.b.1153.1		2
16.5	even	4		2304.2.d.r.1153.2		2
16.11	odd	4		2304.2.d.b.1153.2		2
16.13	even	4		2304.2.d.r.1153.1		2
21.20	even	2		6272.2.a.h.1.1		1
24.5	odd	2		128.2.a.d.1.1	yes	1
24.11	even	2		128.2.a.b.1.1	yes	1
48.5	odd	4		256.2.b.c.129.2		2
48.11	even	4		256.2.b.a.129.1		2
48.29	odd	4		256.2.b.c.129.1		2
48.35	even	4		256.2.b.a.129.2		2
60.23	odd	4		3200.2.c.f.2049.2		2
60.47	odd	4		3200.2.c.f.2049.1		2
60.59	even	2		3200.2.a.e.1.1		1
84.83	odd	2		6272.2.a.b.1.1		1
96.5	odd	8		1024.2.e.i.769.2		4
96.11	even	8		1024.2.e.m.769.2		4
96.29	odd	8		1024.2.e.i.257.1		4
96.35	even	8		1024.2.e.m.257.2		4
96.53	odd	8		1024.2.e.i.769.1		4
96.59	even	8		1024.2.e.m.769.1		4
96.77	odd	8		1024.2.e.i.257.2		4
96.83	even	8		1024.2.e.m.257.1		4
120.29	odd	2		3200.2.a.h.1.1		1
120.53	even	4		3200.2.c.e.2049.2		2
120.59	even	2		3200.2.a.u.1.1		1
120.77	even	4		3200.2.c.e.2049.1		2
120.83	odd	4		3200.2.c.k.2049.1		2
120.107	odd	4		3200.2.c.k.2049.2		2
168.83	odd	2		6272.2.a.g.1.1		1
168.125	even	2		6272.2.a.a.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.2.a.a.1.1	✓	1	3.2	odd	2
128.2.a.b.1.1	yes	1	24.11	even	2
128.2.a.c.1.1	yes	1	12.11	even	2
128.2.a.d.1.1	yes	1	24.5	odd	2
256.2.b.a.129.1		2	48.11	even	4
256.2.b.a.129.2		2	48.35	even	4
256.2.b.c.129.1		2	48.29	odd	4
256.2.b.c.129.2		2	48.5	odd	4
1024.2.e.i.257.1		4	96.29	odd	8
1024.2.e.i.257.2		4	96.77	odd	8
1024.2.e.i.769.1		4	96.53	odd	8
1024.2.e.i.769.2		4	96.5	odd	8
1024.2.e.m.257.1		4	96.83	even	8
1024.2.e.m.257.2		4	96.35	even	8
1024.2.e.m.769.1		4	96.59	even	8
1024.2.e.m.769.2		4	96.11	even	8
1152.2.a.c.1.1		1	8.5	even	2
1152.2.a.h.1.1		1	8.3	odd	2
1152.2.a.m.1.1		1	1.1	even	1	trivial
1152.2.a.r.1.1		1	4.3	odd	2
2304.2.d.b.1153.1		2	16.3	odd	4
2304.2.d.b.1153.2		2	16.11	odd	4
2304.2.d.r.1153.1		2	16.13	even	4
2304.2.d.r.1153.2		2	16.5	even	4
3200.2.a.e.1.1		1	60.59	even	2
3200.2.a.h.1.1		1	120.29	odd	2
3200.2.a.u.1.1		1	120.59	even	2
3200.2.a.x.1.1		1	15.14	odd	2
3200.2.c.e.2049.1		2	120.77	even	4
3200.2.c.e.2049.2		2	120.53	even	4
3200.2.c.f.2049.1		2	60.47	odd	4
3200.2.c.f.2049.2		2	60.23	odd	4
3200.2.c.k.2049.1		2	120.83	odd	4
3200.2.c.k.2049.2		2	120.107	odd	4
3200.2.c.l.2049.1		2	15.8	even	4
3200.2.c.l.2049.2		2	15.2	even	4
6272.2.a.a.1.1		1	168.125	even	2
6272.2.a.b.1.1		1	84.83	odd	2
6272.2.a.g.1.1		1	168.83	odd	2
6272.2.a.h.1.1		1	21.20	even	2