Embedded newform 648.1.b.a.163.1

• Label: 648.1.b.a.163.1
• Level: $648$
• Weight: $1$
• Character: 648.163
• Self dual: yes
• Analytic conductor: $0.323$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{3}$
• CM discriminant: -8
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	648.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.323394128186\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 72)
Projective image:	\(D_{3}\)
Projective field:	Galois closure of 3.1.648.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.0.1259712.1
Stark unit:	Root of $x^{6} - 24x^{5} + 114x^{4} - 398x^{3} + 114x^{2} - 24x + 1$

Embedding invariants

Embedding label			163.1
Character	\(\chi\)	\(=\)	648.163

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{8} +O(q^{10})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	−1.00000
			\(3\)	0	0
\(5\)	0	0				1.00000			\(0\)
			−1.00000			\(\pi\)
\(7\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(11\)	1.00000	1.00000	0.500000	−	0.866025i	\(-0.333333\pi\)
			0.500000	+	0.866025i	\(0.333333\pi\)
\(13\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(17\)	1.00000	1.00000	0.500000	−	0.866025i	\(-0.333333\pi\)
			0.500000	+	0.866025i	\(0.333333\pi\)
\(19\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(23\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(29\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(31\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(37\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(41\)	1.00000	1.00000	0.500000	−	0.866025i	\(-0.333333\pi\)
			0.500000	+	0.866025i	\(0.333333\pi\)
\(43\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(47\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.1.b.a.163.1		1
3.2	odd	2		648.1.b.b.163.1		1
4.3	odd	2		2592.1.b.a.1135.1		1
8.3	odd	2	CM	648.1.b.a.163.1		1
8.5	even	2		2592.1.b.a.1135.1		1
9.2	odd	6		72.1.p.a.67.1	yes	2
9.4	even	3		216.1.p.a.19.1		2
9.5	odd	6		72.1.p.a.43.1	✓	2
9.7	even	3		216.1.p.a.91.1		2
12.11	even	2		2592.1.b.b.1135.1		1
24.5	odd	2		2592.1.b.b.1135.1		1
24.11	even	2		648.1.b.b.163.1		1
36.7	odd	6		864.1.t.a.847.1		2
36.11	even	6		288.1.t.a.175.1		2
36.23	even	6		288.1.t.a.79.1		2
36.31	odd	6		864.1.t.a.559.1		2
45.2	even	12		1800.1.ba.b.499.2		4
45.14	odd	6		1800.1.bk.d.1051.1		2
45.23	even	12		1800.1.ba.b.1699.2		4
45.29	odd	6		1800.1.bk.d.1651.1		2
45.32	even	12		1800.1.ba.b.1699.1		4
45.38	even	12		1800.1.ba.b.499.1		4
63.2	odd	6		3528.1.ba.b.67.1		2
63.5	even	6		3528.1.ce.b.2419.1		2
63.11	odd	6		3528.1.ce.a.3019.1		2
63.20	even	6		3528.1.cg.a.3235.1		2
63.23	odd	6		3528.1.ce.a.2419.1		2
63.32	odd	6		3528.1.ba.b.1843.1		2
63.38	even	6		3528.1.ce.b.3019.1		2
63.41	even	6		3528.1.cg.a.2059.1		2
63.47	even	6		3528.1.ba.a.67.1		2
63.59	even	6		3528.1.ba.a.1843.1		2
72.5	odd	6		288.1.t.a.79.1		2
72.11	even	6		72.1.p.a.67.1	yes	2
72.13	even	6		864.1.t.a.559.1		2
72.29	odd	6		288.1.t.a.175.1		2
72.43	odd	6		216.1.p.a.91.1		2
72.59	even	6		72.1.p.a.43.1	✓	2
72.61	even	6		864.1.t.a.847.1		2
72.67	odd	6		216.1.p.a.19.1		2
144.5	odd	12		2304.1.o.c.511.2		4
144.11	even	12		2304.1.o.c.2047.2		4
144.29	odd	12		2304.1.o.c.2047.2		4
144.59	even	12		2304.1.o.c.511.1		4
144.77	odd	12		2304.1.o.c.511.1		4
144.83	even	12		2304.1.o.c.2047.1		4
144.101	odd	12		2304.1.o.c.2047.1		4
144.131	even	12		2304.1.o.c.511.2		4
360.59	even	6		1800.1.bk.d.1051.1		2
360.83	odd	12		1800.1.ba.b.499.1		4
360.203	odd	12		1800.1.ba.b.1699.2		4
360.227	odd	12		1800.1.ba.b.499.2		4
360.299	even	6		1800.1.bk.d.1651.1		2
360.347	odd	12		1800.1.ba.b.1699.1		4
504.11	even	6		3528.1.ce.a.3019.1		2
504.59	odd	6		3528.1.ba.a.1843.1		2
504.83	odd	6		3528.1.cg.a.3235.1		2
504.131	odd	6		3528.1.ce.b.2419.1		2
504.227	odd	6		3528.1.ce.b.3019.1		2
504.275	even	6		3528.1.ce.a.2419.1		2
504.299	odd	6		3528.1.ba.a.67.1		2
504.347	even	6		3528.1.ba.b.1843.1		2
504.419	odd	6		3528.1.cg.a.2059.1		2
504.443	even	6		3528.1.ba.b.67.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.1.p.a.43.1	✓	2	9.5	odd	6
72.1.p.a.43.1	✓	2	72.59	even	6
72.1.p.a.67.1	yes	2	9.2	odd	6
72.1.p.a.67.1	yes	2	72.11	even	6
216.1.p.a.19.1		2	9.4	even	3
216.1.p.a.19.1		2	72.67	odd	6
216.1.p.a.91.1		2	9.7	even	3
216.1.p.a.91.1		2	72.43	odd	6
288.1.t.a.79.1		2	36.23	even	6
288.1.t.a.79.1		2	72.5	odd	6
288.1.t.a.175.1		2	36.11	even	6
288.1.t.a.175.1		2	72.29	odd	6
648.1.b.a.163.1		1	1.1	even	1	trivial
648.1.b.a.163.1		1	8.3	odd	2	CM
648.1.b.b.163.1		1	3.2	odd	2
648.1.b.b.163.1		1	24.11	even	2
864.1.t.a.559.1		2	36.31	odd	6
864.1.t.a.559.1		2	72.13	even	6
864.1.t.a.847.1		2	36.7	odd	6
864.1.t.a.847.1		2	72.61	even	6
1800.1.ba.b.499.1		4	45.38	even	12
1800.1.ba.b.499.1		4	360.83	odd	12
1800.1.ba.b.499.2		4	45.2	even	12
1800.1.ba.b.499.2		4	360.227	odd	12
1800.1.ba.b.1699.1		4	45.32	even	12
1800.1.ba.b.1699.1		4	360.347	odd	12
1800.1.ba.b.1699.2		4	45.23	even	12
1800.1.ba.b.1699.2		4	360.203	odd	12
1800.1.bk.d.1051.1		2	45.14	odd	6
1800.1.bk.d.1051.1		2	360.59	even	6
1800.1.bk.d.1651.1		2	45.29	odd	6
1800.1.bk.d.1651.1		2	360.299	even	6
2304.1.o.c.511.1		4	144.59	even	12
2304.1.o.c.511.1		4	144.77	odd	12
2304.1.o.c.511.2		4	144.5	odd	12
2304.1.o.c.511.2		4	144.131	even	12
2304.1.o.c.2047.1		4	144.83	even	12
2304.1.o.c.2047.1		4	144.101	odd	12
2304.1.o.c.2047.2		4	144.11	even	12
2304.1.o.c.2047.2		4	144.29	odd	12
2592.1.b.a.1135.1		1	4.3	odd	2
2592.1.b.a.1135.1		1	8.5	even	2
2592.1.b.b.1135.1		1	12.11	even	2
2592.1.b.b.1135.1		1	24.5	odd	2
3528.1.ba.a.67.1		2	63.47	even	6
3528.1.ba.a.67.1		2	504.299	odd	6
3528.1.ba.a.1843.1		2	63.59	even	6
3528.1.ba.a.1843.1		2	504.59	odd	6
3528.1.ba.b.67.1		2	63.2	odd	6
3528.1.ba.b.67.1		2	504.443	even	6
3528.1.ba.b.1843.1		2	63.32	odd	6
3528.1.ba.b.1843.1		2	504.347	even	6
3528.1.ce.a.2419.1		2	63.23	odd	6
3528.1.ce.a.2419.1		2	504.275	even	6
3528.1.ce.a.3019.1		2	63.11	odd	6
3528.1.ce.a.3019.1		2	504.11	even	6
3528.1.ce.b.2419.1		2	63.5	even	6
3528.1.ce.b.2419.1		2	504.131	odd	6
3528.1.ce.b.3019.1		2	63.38	even	6
3528.1.ce.b.3019.1		2	504.227	odd	6
3528.1.cg.a.2059.1		2	63.41	even	6
3528.1.cg.a.2059.1		2	504.419	odd	6
3528.1.cg.a.3235.1		2	63.20	even	6
3528.1.cg.a.3235.1		2	504.83	odd	6