Embedded newform 588.4.a.c.1.1

• Label: 588.4.a.c.1.1
• Level: $588$
• Weight: $4$
• Character: 588.1
• Self dual: yes
• Analytic conductor: $34.693$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	588.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(34.6931230834\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 12)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	588.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-3.00000 q^{3} +18.0000 q^{5} +9.00000 q^{9} +36.0000 q^{11} +10.0000 q^{13} -54.0000 q^{15} -18.0000 q^{17} +100.000 q^{19} +72.0000 q^{23} +199.000 q^{25} -27.0000 q^{27} -234.000 q^{29} +16.0000 q^{31} -108.000 q^{33} -226.000 q^{37} -30.0000 q^{39} -90.0000 q^{41} +452.000 q^{43} +162.000 q^{45} -432.000 q^{47} +54.0000 q^{51} +414.000 q^{53} +648.000 q^{55} -300.000 q^{57} +684.000 q^{59} -422.000 q^{61} +180.000 q^{65} +332.000 q^{67} -216.000 q^{69} -360.000 q^{71} -26.0000 q^{73} -597.000 q^{75} +512.000 q^{79} +81.0000 q^{81} +1188.00 q^{83} -324.000 q^{85} +702.000 q^{87} +630.000 q^{89} -48.0000 q^{93} +1800.00 q^{95} +1054.00 q^{97} +324.000 q^{99} +O(q^{100})\)

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
			\(3\)	−3.00000	−0.577350
\(5\)	18.0000	1.60997				0.804984	−	0.593296i	\(-0.202174\pi\)
			0.804984	+	0.593296i	\(0.202174\pi\)
\(7\)	0	0
			\(11\)	36.0000	0.986764	0.493382	−	0.869813i	\(-0.335760\pi\)
0.493382	+	0.869813i				\(0.335760\pi\)
\(13\)	10.0000	0.213346	0.106673	−	0.994294i	\(-0.465980\pi\)
			0.106673	+	0.994294i	\(0.465980\pi\)
\(17\)	−18.0000	−0.256802	−0.128401	−	0.991722i	\(-0.540985\pi\)
			−0.128401	+	0.991722i	\(0.540985\pi\)
\(19\)	100.000	1.20745	0.603726	−	0.797192i	\(-0.293682\pi\)
			0.603726	+	0.797192i	\(0.293682\pi\)
\(23\)	72.0000	0.652741	0.326370	−	0.945242i	\(-0.394174\pi\)
			0.326370	+	0.945242i	\(0.394174\pi\)
\(29\)	−234.000	−1.49837	−0.749185	−	0.662361i	\(-0.769554\pi\)
			−0.749185	+	0.662361i	\(0.769554\pi\)
\(31\)	16.0000	0.0926995	0.0463498	−	0.998925i	\(-0.485241\pi\)
			0.0463498	+	0.998925i	\(0.485241\pi\)
\(37\)	−226.000	−1.00417	−0.502083	−	0.864819i	\(-0.667433\pi\)
			−0.502083	+	0.864819i	\(0.667433\pi\)
\(41\)	−90.0000	−0.342820	−0.171410	−	0.985200i	\(-0.554832\pi\)
			−0.171410	+	0.985200i	\(0.554832\pi\)
\(43\)	452.000	1.60301	0.801504	−	0.597989i	\(-0.204033\pi\)
			0.801504	+	0.597989i	\(0.204033\pi\)
\(47\)	−432.000	−1.34072	−0.670358	−	0.742038i	\(-0.733860\pi\)
			−0.670358	+	0.742038i	\(0.733860\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.4.a.c.1.1		1
3.2	odd	2		1764.4.a.b.1.1		1
4.3	odd	2		2352.4.a.bk.1.1		1
7.2	even	3		588.4.i.e.361.1		2
7.3	odd	6		588.4.i.d.373.1		2
7.4	even	3		588.4.i.e.373.1		2
7.5	odd	6		588.4.i.d.361.1		2
7.6	odd	2		12.4.a.a.1.1	✓	1
21.2	odd	6		1764.4.k.o.361.1		2
21.5	even	6		1764.4.k.b.361.1		2
21.11	odd	6		1764.4.k.o.1549.1		2
21.17	even	6		1764.4.k.b.1549.1		2
21.20	even	2		36.4.a.a.1.1		1
28.27	even	2		48.4.a.a.1.1		1
35.13	even	4		300.4.d.e.49.2		2
35.27	even	4		300.4.d.e.49.1		2
35.34	odd	2		300.4.a.b.1.1		1
56.13	odd	2		192.4.a.f.1.1		1
56.27	even	2		192.4.a.l.1.1		1
63.13	odd	6		324.4.e.h.217.1		2
63.20	even	6		324.4.e.a.109.1		2
63.34	odd	6		324.4.e.h.109.1		2
63.41	even	6		324.4.e.a.217.1		2
77.76	even	2		1452.4.a.d.1.1		1
84.83	odd	2		144.4.a.g.1.1		1
91.34	even	4		2028.4.b.c.337.1		2
91.83	even	4		2028.4.b.c.337.2		2
91.90	odd	2		2028.4.a.c.1.1		1
105.62	odd	4		900.4.d.c.649.2		2
105.83	odd	4		900.4.d.c.649.1		2
105.104	even	2		900.4.a.g.1.1		1
112.13	odd	4		768.4.d.g.385.2		2
112.27	even	4		768.4.d.j.385.2		2
112.69	odd	4		768.4.d.g.385.1		2
112.83	even	4		768.4.d.j.385.1		2
140.27	odd	4		1200.4.f.d.49.2		2
140.83	odd	4		1200.4.f.d.49.1		2
140.139	even	2		1200.4.a.be.1.1		1
168.83	odd	2		576.4.a.a.1.1		1
168.125	even	2		576.4.a.b.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.4.a.a.1.1	✓	1	7.6	odd	2
36.4.a.a.1.1		1	21.20	even	2
48.4.a.a.1.1		1	28.27	even	2
144.4.a.g.1.1		1	84.83	odd	2
192.4.a.f.1.1		1	56.13	odd	2
192.4.a.l.1.1		1	56.27	even	2
300.4.a.b.1.1		1	35.34	odd	2
300.4.d.e.49.1		2	35.27	even	4
300.4.d.e.49.2		2	35.13	even	4
324.4.e.a.109.1		2	63.20	even	6
324.4.e.a.217.1		2	63.41	even	6
324.4.e.h.109.1		2	63.34	odd	6
324.4.e.h.217.1		2	63.13	odd	6
576.4.a.a.1.1		1	168.83	odd	2
576.4.a.b.1.1		1	168.125	even	2
588.4.a.c.1.1		1	1.1	even	1	trivial
588.4.i.d.361.1		2	7.5	odd	6
588.4.i.d.373.1		2	7.3	odd	6
588.4.i.e.361.1		2	7.2	even	3
588.4.i.e.373.1		2	7.4	even	3
768.4.d.g.385.1		2	112.69	odd	4
768.4.d.g.385.2		2	112.13	odd	4
768.4.d.j.385.1		2	112.83	even	4
768.4.d.j.385.2		2	112.27	even	4
900.4.a.g.1.1		1	105.104	even	2
900.4.d.c.649.1		2	105.83	odd	4
900.4.d.c.649.2		2	105.62	odd	4
1200.4.a.be.1.1		1	140.139	even	2
1200.4.f.d.49.1		2	140.83	odd	4
1200.4.f.d.49.2		2	140.27	odd	4
1452.4.a.d.1.1		1	77.76	even	2
1764.4.a.b.1.1		1	3.2	odd	2
1764.4.k.b.361.1		2	21.5	even	6
1764.4.k.b.1549.1		2	21.17	even	6
1764.4.k.o.361.1		2	21.2	odd	6
1764.4.k.o.1549.1		2	21.11	odd	6
2028.4.a.c.1.1		1	91.90	odd	2
2028.4.b.c.337.1		2	91.34	even	4
2028.4.b.c.337.2		2	91.83	even	4
2352.4.a.bk.1.1		1	4.3	odd	2