Embedded newform 1470.2.a.b.1.1

• Label: 1470.2.a.b.1.1
• Level: $1470$
• Weight: $2$
• Character: 1470.1
• Self dual: yes
• Analytic conductor: $11.738$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1470.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(11.7380090971\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 210)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1470.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)

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	−0.707107
			\(3\)	−1.00000	−0.577350
\(5\)	−1.00000	−0.447214
			\(7\)	0	0
\(11\)	0	0					−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(13\)	−2.00000	−0.554700	−0.277350	−	0.960769i	\(-0.589456\pi\)
			−0.277350	+	0.960769i	\(0.589456\pi\)
\(17\)	6.00000	1.45521	0.727607	−	0.685994i	\(-0.240633\pi\)
			0.727607	+	0.685994i	\(0.240633\pi\)
\(19\)	−8.00000	−1.83533	−0.917663	−	0.397360i	\(-0.869927\pi\)
			−0.917663	+	0.397360i	\(0.869927\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	6.00000	1.11417	0.557086	−	0.830455i	\(-0.311919\pi\)
			0.557086	+	0.830455i	\(0.311919\pi\)
\(31\)	4.00000	0.718421	0.359211	−	0.933257i	\(-0.383046\pi\)
			0.359211	+	0.933257i	\(0.383046\pi\)
\(37\)	−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	\(0.344787\pi\)
\(43\)	−4.00000	−0.609994	−0.304997	−	0.952353i	\(-0.598656\pi\)
			−0.304997	+	0.952353i	\(0.598656\pi\)
\(47\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1470.2.a.b.1.1		1
3.2	odd	2		4410.2.a.bi.1.1		1
5.4	even	2		7350.2.a.cs.1.1		1
7.2	even	3		1470.2.i.s.361.1		2
7.3	odd	6		1470.2.i.l.961.1		2
7.4	even	3		1470.2.i.s.961.1		2
7.5	odd	6		1470.2.i.l.361.1		2
7.6	odd	2		210.2.a.b.1.1	✓	1
21.20	even	2		630.2.a.h.1.1		1
28.27	even	2		1680.2.a.g.1.1		1
35.13	even	4		1050.2.g.c.799.2		2
35.27	even	4		1050.2.g.c.799.1		2
35.34	odd	2		1050.2.a.k.1.1		1
56.13	odd	2		6720.2.a.n.1.1		1
56.27	even	2		6720.2.a.bi.1.1		1
84.83	odd	2		5040.2.a.g.1.1		1
105.62	odd	4		3150.2.g.i.2899.2		2
105.83	odd	4		3150.2.g.i.2899.1		2
105.104	even	2		3150.2.a.f.1.1		1
140.139	even	2		8400.2.a.cm.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.a.b.1.1	✓	1	7.6	odd	2
630.2.a.h.1.1		1	21.20	even	2
1050.2.a.k.1.1		1	35.34	odd	2
1050.2.g.c.799.1		2	35.27	even	4
1050.2.g.c.799.2		2	35.13	even	4
1470.2.a.b.1.1		1	1.1	even	1	trivial
1470.2.i.l.361.1		2	7.5	odd	6
1470.2.i.l.961.1		2	7.3	odd	6
1470.2.i.s.361.1		2	7.2	even	3
1470.2.i.s.961.1		2	7.4	even	3
1680.2.a.g.1.1		1	28.27	even	2
3150.2.a.f.1.1		1	105.104	even	2
3150.2.g.i.2899.1		2	105.83	odd	4
3150.2.g.i.2899.2		2	105.62	odd	4
4410.2.a.bi.1.1		1	3.2	odd	2
5040.2.a.g.1.1		1	84.83	odd	2
6720.2.a.n.1.1		1	56.13	odd	2
6720.2.a.bi.1.1		1	56.27	even	2
7350.2.a.cs.1.1		1	5.4	even	2
8400.2.a.cm.1.1		1	140.139	even	2