Embedded newform 165.1.l.a.32.1

• Label: 165.1.l.a.32.1
• Level: $165$
• Weight: $1$
• Character: 165.32
• Analytic conductor: $0.082$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{4}$
• CM discriminant: -11
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	165.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.0823457270844\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{4}\)
Projective field:	Galois closure of 4.2.12375.1
Artin image:	$C_4\wr C_2$
Artin field:	Galois closure of 8.0.16471125.2

Embedding invariants

Embedding label			32.1
Root			\(-1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	165.32
Dual form			165.1.l.a.98.1

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000i q^{4} -1.00000i q^{5} +1.00000 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/165\mathbb{Z}\right)^\times\).

\(n\)	\(46\)	\(56\)	\(67\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)

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			0		0.707107	−	0.707107i	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(3\)	−1.00000			−1.00000
							\(5\)		−	1.00000i		−	1.00000i
\(7\)		0			0									−0.707107	−	0.707107i	\(-0.750000\pi\)
							0.707107	+	0.707107i	\(0.250000\pi\)
\(11\)			1.00000i			1.00000i
							\(13\)		0			0		0.707107	−	0.707107i	\(-0.250000\pi\)
−0.707107	+	0.707107i	\(0.750000\pi\)
\(17\)		0			0		0.707107	−	0.707107i	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(19\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(23\)	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			\(0\)
									1.00000i	\(0.5\pi\)
\(29\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(31\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(37\)	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			\(0\)
									1.00000i	\(0.5\pi\)
\(41\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(43\)		0			0		0.707107	−	0.707107i	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(47\)	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	\(-0.5\pi\)
							1.00000			\(0\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	165.1.l.a.32.1	✓	2
3.2	odd	2		165.1.l.b.32.1	yes	2
4.3	odd	2		2640.1.ch.b.2177.1		2
5.2	odd	4		825.1.l.a.593.1		2
5.3	odd	4		165.1.l.b.98.1	yes	2
5.4	even	2		825.1.l.b.32.1		2
11.2	odd	10		1815.1.v.b.887.1		8
11.3	even	5		1815.1.v.b.1322.1		8
11.4	even	5		1815.1.v.b.1667.1		8
11.5	even	5		1815.1.v.b.602.1		8
11.6	odd	10		1815.1.v.b.602.1		8
11.7	odd	10		1815.1.v.b.1667.1		8
11.8	odd	10		1815.1.v.b.1322.1		8
11.9	even	5		1815.1.v.b.887.1		8
11.10	odd	2	CM	165.1.l.a.32.1	✓	2
12.11	even	2		2640.1.ch.a.2177.1		2
15.2	even	4		825.1.l.b.593.1		2
15.8	even	4	inner	165.1.l.a.98.1	yes	2
15.14	odd	2		825.1.l.a.32.1		2
20.3	even	4		2640.1.ch.a.593.1		2
33.2	even	10		1815.1.v.a.887.1		8
33.5	odd	10		1815.1.v.a.602.1		8
33.8	even	10		1815.1.v.a.1322.1		8
33.14	odd	10		1815.1.v.a.1322.1		8
33.17	even	10		1815.1.v.a.602.1		8
33.20	odd	10		1815.1.v.a.887.1		8
33.26	odd	10		1815.1.v.a.1667.1		8
33.29	even	10		1815.1.v.a.1667.1		8
33.32	even	2		165.1.l.b.32.1	yes	2
44.43	even	2		2640.1.ch.b.2177.1		2
55.3	odd	20		1815.1.v.a.233.1		8
55.8	even	20		1815.1.v.a.233.1		8
55.13	even	20		1815.1.v.a.1613.1		8
55.18	even	20		1815.1.v.a.578.1		8
55.28	even	20		1815.1.v.a.1328.1		8
55.32	even	4		825.1.l.a.593.1		2
55.38	odd	20		1815.1.v.a.1328.1		8
55.43	even	4		165.1.l.b.98.1	yes	2
55.48	odd	20		1815.1.v.a.578.1		8
55.53	odd	20		1815.1.v.a.1613.1		8
55.54	odd	2		825.1.l.b.32.1		2
60.23	odd	4		2640.1.ch.b.593.1		2
132.131	odd	2		2640.1.ch.a.2177.1		2
165.8	odd	20		1815.1.v.b.233.1		8
165.32	odd	4		825.1.l.b.593.1		2
165.38	even	20		1815.1.v.b.1328.1		8
165.53	even	20		1815.1.v.b.1613.1		8
165.68	odd	20		1815.1.v.b.1613.1		8
165.83	odd	20		1815.1.v.b.1328.1		8
165.98	odd	4	inner	165.1.l.a.98.1	yes	2
165.113	even	20		1815.1.v.b.233.1		8
165.128	odd	20		1815.1.v.b.578.1		8
165.158	even	20		1815.1.v.b.578.1		8
165.164	even	2		825.1.l.a.32.1		2
220.43	odd	4		2640.1.ch.a.593.1		2
660.263	even	4		2640.1.ch.b.593.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.1.l.a.32.1	✓	2	1.1	even	1	trivial
165.1.l.a.32.1	✓	2	11.10	odd	2	CM
165.1.l.a.98.1	yes	2	15.8	even	4	inner
165.1.l.a.98.1	yes	2	165.98	odd	4	inner
165.1.l.b.32.1	yes	2	3.2	odd	2
165.1.l.b.32.1	yes	2	33.32	even	2
165.1.l.b.98.1	yes	2	5.3	odd	4
165.1.l.b.98.1	yes	2	55.43	even	4
825.1.l.a.32.1		2	15.14	odd	2
825.1.l.a.32.1		2	165.164	even	2
825.1.l.a.593.1		2	5.2	odd	4
825.1.l.a.593.1		2	55.32	even	4
825.1.l.b.32.1		2	5.4	even	2
825.1.l.b.32.1		2	55.54	odd	2
825.1.l.b.593.1		2	15.2	even	4
825.1.l.b.593.1		2	165.32	odd	4
1815.1.v.a.233.1		8	55.3	odd	20
1815.1.v.a.233.1		8	55.8	even	20
1815.1.v.a.578.1		8	55.18	even	20
1815.1.v.a.578.1		8	55.48	odd	20
1815.1.v.a.602.1		8	33.5	odd	10
1815.1.v.a.602.1		8	33.17	even	10
1815.1.v.a.887.1		8	33.2	even	10
1815.1.v.a.887.1		8	33.20	odd	10
1815.1.v.a.1322.1		8	33.8	even	10
1815.1.v.a.1322.1		8	33.14	odd	10
1815.1.v.a.1328.1		8	55.28	even	20
1815.1.v.a.1328.1		8	55.38	odd	20
1815.1.v.a.1613.1		8	55.13	even	20
1815.1.v.a.1613.1		8	55.53	odd	20
1815.1.v.a.1667.1		8	33.26	odd	10
1815.1.v.a.1667.1		8	33.29	even	10
1815.1.v.b.233.1		8	165.8	odd	20
1815.1.v.b.233.1		8	165.113	even	20
1815.1.v.b.578.1		8	165.128	odd	20
1815.1.v.b.578.1		8	165.158	even	20
1815.1.v.b.602.1		8	11.5	even	5
1815.1.v.b.602.1		8	11.6	odd	10
1815.1.v.b.887.1		8	11.2	odd	10
1815.1.v.b.887.1		8	11.9	even	5
1815.1.v.b.1322.1		8	11.3	even	5
1815.1.v.b.1322.1		8	11.8	odd	10
1815.1.v.b.1328.1		8	165.38	even	20
1815.1.v.b.1328.1		8	165.83	odd	20
1815.1.v.b.1613.1		8	165.53	even	20
1815.1.v.b.1613.1		8	165.68	odd	20
1815.1.v.b.1667.1		8	11.4	even	5
1815.1.v.b.1667.1		8	11.7	odd	10
2640.1.ch.a.593.1		2	20.3	even	4
2640.1.ch.a.593.1		2	220.43	odd	4
2640.1.ch.a.2177.1		2	12.11	even	2
2640.1.ch.a.2177.1		2	132.131	odd	2
2640.1.ch.b.593.1		2	60.23	odd	4
2640.1.ch.b.593.1		2	660.263	even	4
2640.1.ch.b.2177.1		2	4.3	odd	2
2640.1.ch.b.2177.1		2	44.43	even	2