Embedded newform 1568.2.i.g.961.1

• Label: 1568.2.i.g.961.1
• Level: $1568$
• Weight: $2$
• Character: 1568.961
• Analytic conductor: $12.521$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -4
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.5205430369\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 32)
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label			961.1
Root			\(0.500000 - 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	1568.961
Dual form			1568.2.i.g.1537.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{5} +(1.50000 - 2.59808i) q^{9} +6.00000 q^{13} +(-1.00000 - 1.73205i) q^{17} +(0.500000 + 0.866025i) q^{25} -10.0000 q^{29} +(1.00000 - 1.73205i) q^{37} +10.0000 q^{41} +(-3.00000 - 5.19615i) q^{45} +(-7.00000 - 12.1244i) q^{53} +(5.00000 - 8.66025i) q^{61} +(6.00000 - 10.3923i) q^{65} +(3.00000 + 5.19615i) q^{73} +(-4.50000 - 7.79423i) q^{81} -4.00000 q^{85} +(-5.00000 + 8.66025i) q^{89} +18.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 3 q^{9} + 12 q^{13} - 2 q^{17} + q^{25} - 20 q^{29} + 2 q^{37} + 20 q^{41} - 6 q^{45} - 14 q^{53} + 10 q^{61} + 12 q^{65} + 6 q^{73} - 9 q^{81} - 8 q^{85} - 10 q^{89} + 36 q^{97}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(1471\)	\(1473\)
\(\chi(n)\)	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

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
							\(3\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
−0.866025	+	0.500000i	\(0.833333\pi\)
\(5\)	1.00000	−	1.73205i	0.447214	−	0.774597i	−0.550990	−	0.834512i	\(-0.685750\pi\)
							0.998203	+	0.0599153i	\(0.0190830\pi\)
\(7\)		0			0
							\(11\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
−0.866025	+	0.500000i	\(0.833333\pi\)
\(13\)	6.00000			1.66410			0.832050	−	0.554700i	\(-0.187167\pi\)
							0.832050	+	0.554700i	\(0.187167\pi\)
\(17\)	−1.00000	−	1.73205i	−0.242536	−	0.420084i	0.718900	−	0.695113i	\(-0.244646\pi\)
							−0.961436	+	0.275029i	\(0.911312\pi\)
\(19\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(23\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(29\)	−10.0000			−1.85695			−0.928477	−	0.371391i	\(-0.878881\pi\)
							−0.928477	+	0.371391i	\(0.878881\pi\)
\(31\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(37\)	1.00000	−	1.73205i	0.164399	−	0.284747i	−0.772043	−	0.635571i	\(-0.780765\pi\)
							0.936442	+	0.350823i	\(0.114098\pi\)
\(41\)	10.0000			1.56174			0.780869	−	0.624695i	\(-0.214777\pi\)
							0.780869	+	0.624695i	\(0.214777\pi\)
\(43\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(47\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.2.i.g.961.1		2
4.3	odd	2	CM	1568.2.i.g.961.1		2
7.2	even	3		32.2.a.a.1.1	✓	1
7.3	odd	6		1568.2.i.f.1537.1		2
7.4	even	3	inner	1568.2.i.g.1537.1		2
7.5	odd	6		1568.2.a.e.1.1		1
7.6	odd	2		1568.2.i.f.961.1		2
21.2	odd	6		288.2.a.d.1.1		1
28.3	even	6		1568.2.i.f.1537.1		2
28.11	odd	6	inner	1568.2.i.g.1537.1		2
28.19	even	6		1568.2.a.e.1.1		1
28.23	odd	6		32.2.a.a.1.1	✓	1
28.27	even	2		1568.2.i.f.961.1		2
35.2	odd	12		800.2.c.e.449.1		2
35.9	even	6		800.2.a.d.1.1		1
35.23	odd	12		800.2.c.e.449.2		2
56.5	odd	6		3136.2.a.m.1.1		1
56.19	even	6		3136.2.a.m.1.1		1
56.37	even	6		64.2.a.a.1.1		1
56.51	odd	6		64.2.a.a.1.1		1
63.2	odd	6		2592.2.i.e.1729.1		2
63.16	even	3		2592.2.i.t.1729.1		2
63.23	odd	6		2592.2.i.e.865.1		2
63.58	even	3		2592.2.i.t.865.1		2
77.65	odd	6		3872.2.a.f.1.1		1
84.23	even	6		288.2.a.d.1.1		1
91.51	even	6		5408.2.a.g.1.1		1
105.2	even	12		7200.2.f.m.6049.1		2
105.23	even	12		7200.2.f.m.6049.2		2
105.44	odd	6		7200.2.a.v.1.1		1
112.37	even	12		256.2.b.b.129.1		2
112.51	odd	12		256.2.b.b.129.2		2
112.93	even	12		256.2.b.b.129.2		2
112.107	odd	12		256.2.b.b.129.1		2
119.16	even	6		9248.2.a.f.1.1		1
140.23	even	12		800.2.c.e.449.2		2
140.79	odd	6		800.2.a.d.1.1		1
140.107	even	12		800.2.c.e.449.1		2
168.107	even	6		576.2.a.c.1.1		1
168.149	odd	6		576.2.a.c.1.1		1
224.37	even	24		1024.2.e.j.769.1		4
224.51	odd	24		1024.2.e.j.257.1		4
224.93	even	24		1024.2.e.j.257.2		4
224.107	odd	24		1024.2.e.j.769.2		4
224.149	even	24		1024.2.e.j.769.2		4
224.163	odd	24		1024.2.e.j.257.2		4
224.205	even	24		1024.2.e.j.257.1		4
224.219	odd	24		1024.2.e.j.769.1		4
252.23	even	6		2592.2.i.e.865.1		2
252.79	odd	6		2592.2.i.t.1729.1		2
252.191	even	6		2592.2.i.e.1729.1		2
252.247	odd	6		2592.2.i.t.865.1		2
280.37	odd	12		1600.2.c.l.449.2		2
280.93	odd	12		1600.2.c.l.449.1		2
280.107	even	12		1600.2.c.l.449.2		2
280.149	even	6		1600.2.a.n.1.1		1
280.163	even	12		1600.2.c.l.449.1		2
280.219	odd	6		1600.2.a.n.1.1		1
308.219	even	6		3872.2.a.f.1.1		1
336.107	even	12		2304.2.d.j.1153.2		2
336.149	odd	12		2304.2.d.j.1153.2		2
336.275	even	12		2304.2.d.j.1153.1		2
336.317	odd	12		2304.2.d.j.1153.1		2
364.51	odd	6		5408.2.a.g.1.1		1
420.23	odd	12		7200.2.f.m.6049.2		2
420.107	odd	12		7200.2.f.m.6049.1		2
420.359	even	6		7200.2.a.v.1.1		1
476.135	odd	6		9248.2.a.f.1.1		1
616.219	even	6		7744.2.a.v.1.1		1
616.373	odd	6		7744.2.a.v.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.2.a.a.1.1	✓	1	7.2	even	3
32.2.a.a.1.1	✓	1	28.23	odd	6
64.2.a.a.1.1		1	56.37	even	6
64.2.a.a.1.1		1	56.51	odd	6
256.2.b.b.129.1		2	112.37	even	12
256.2.b.b.129.1		2	112.107	odd	12
256.2.b.b.129.2		2	112.51	odd	12
256.2.b.b.129.2		2	112.93	even	12
288.2.a.d.1.1		1	21.2	odd	6
288.2.a.d.1.1		1	84.23	even	6
576.2.a.c.1.1		1	168.107	even	6
576.2.a.c.1.1		1	168.149	odd	6
800.2.a.d.1.1		1	35.9	even	6
800.2.a.d.1.1		1	140.79	odd	6
800.2.c.e.449.1		2	35.2	odd	12
800.2.c.e.449.1		2	140.107	even	12
800.2.c.e.449.2		2	35.23	odd	12
800.2.c.e.449.2		2	140.23	even	12
1024.2.e.j.257.1		4	224.51	odd	24
1024.2.e.j.257.1		4	224.205	even	24
1024.2.e.j.257.2		4	224.93	even	24
1024.2.e.j.257.2		4	224.163	odd	24
1024.2.e.j.769.1		4	224.37	even	24
1024.2.e.j.769.1		4	224.219	odd	24
1024.2.e.j.769.2		4	224.107	odd	24
1024.2.e.j.769.2		4	224.149	even	24
1568.2.a.e.1.1		1	7.5	odd	6
1568.2.a.e.1.1		1	28.19	even	6
1568.2.i.f.961.1		2	7.6	odd	2
1568.2.i.f.961.1		2	28.27	even	2
1568.2.i.f.1537.1		2	7.3	odd	6
1568.2.i.f.1537.1		2	28.3	even	6
1568.2.i.g.961.1		2	1.1	even	1	trivial
1568.2.i.g.961.1		2	4.3	odd	2	CM
1568.2.i.g.1537.1		2	7.4	even	3	inner
1568.2.i.g.1537.1		2	28.11	odd	6	inner
1600.2.a.n.1.1		1	280.149	even	6
1600.2.a.n.1.1		1	280.219	odd	6
1600.2.c.l.449.1		2	280.93	odd	12
1600.2.c.l.449.1		2	280.163	even	12
1600.2.c.l.449.2		2	280.37	odd	12
1600.2.c.l.449.2		2	280.107	even	12
2304.2.d.j.1153.1		2	336.275	even	12
2304.2.d.j.1153.1		2	336.317	odd	12
2304.2.d.j.1153.2		2	336.107	even	12
2304.2.d.j.1153.2		2	336.149	odd	12
2592.2.i.e.865.1		2	63.23	odd	6
2592.2.i.e.865.1		2	252.23	even	6
2592.2.i.e.1729.1		2	63.2	odd	6
2592.2.i.e.1729.1		2	252.191	even	6
2592.2.i.t.865.1		2	63.58	even	3
2592.2.i.t.865.1		2	252.247	odd	6
2592.2.i.t.1729.1		2	63.16	even	3
2592.2.i.t.1729.1		2	252.79	odd	6
3136.2.a.m.1.1		1	56.5	odd	6
3136.2.a.m.1.1		1	56.19	even	6
3872.2.a.f.1.1		1	77.65	odd	6
3872.2.a.f.1.1		1	308.219	even	6
5408.2.a.g.1.1		1	91.51	even	6
5408.2.a.g.1.1		1	364.51	odd	6
7200.2.a.v.1.1		1	105.44	odd	6
7200.2.a.v.1.1		1	420.359	even	6
7200.2.f.m.6049.1		2	105.2	even	12
7200.2.f.m.6049.1		2	420.107	odd	12
7200.2.f.m.6049.2		2	105.23	even	12
7200.2.f.m.6049.2		2	420.23	odd	12
7744.2.a.v.1.1		1	616.219	even	6
7744.2.a.v.1.1		1	616.373	odd	6
9248.2.a.f.1.1		1	119.16	even	6
9248.2.a.f.1.1		1	476.135	odd	6