Embedded newform 140.2.g.c.111.7

• Label: 140.2.g.c.111.7
• Level: $140$
• Weight: $2$
• Character: 140.111
• Analytic conductor: $1.118$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	140.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.11790562830\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.342102016.5
Defining polynomial:	\( x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			111.7
Root			\(-0.599676 - 1.28078i\) of defining polynomial
Character	\(\chi\)	\(=\)	140.111
Dual form			140.2.g.c.111.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.28078 + 0.599676i) q^{2} -0.936426 q^{3} +(1.28078 + 1.53610i) q^{4} +1.00000i q^{5} +(-1.19935 - 0.561553i) q^{6} +(2.60399 - 0.468213i) q^{7} +(0.719224 + 2.73546i) q^{8} -2.12311 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} + 2 q^{4} + 14 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/140\mathbb{Z}\right)^\times\).

\(n\)	\(57\)	\(71\)	\(101\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.28078	+	0.599676i	0.905646	+	0.424035i
							\(3\)	−0.936426			−0.540646			−0.270323	−	0.962770i	\(-0.587130\pi\)
−0.270323	+	0.962770i	\(0.587130\pi\)
\(5\)			1.00000i			0.447214i
							\(7\)	2.60399	−	0.468213i	0.984217	−	0.176968i
\(11\)		−	2.39871i		−	0.723237i								−0.932326	−	0.361618i	\(-0.882224\pi\)
							0.932326	−	0.361618i	\(-0.117776\pi\)
\(13\)			2.00000i			0.554700i	0.960769	+	0.277350i	\(0.0894562\pi\)
							−0.960769	+	0.277350i	\(0.910544\pi\)
\(17\)		−	7.12311i		−	1.72761i	−0.503829	−	0.863803i	\(-0.668076\pi\)
							0.503829	−	0.863803i	\(-0.331924\pi\)
\(19\)	−2.39871			−0.550301			−0.275150	−	0.961401i	\(-0.588728\pi\)
							−0.275150	+	0.961401i	\(0.588728\pi\)
\(23\)		−	5.73384i		−	1.19559i	−0.801650	−	0.597794i	\(-0.796044\pi\)
							0.801650	−	0.597794i	\(-0.203956\pi\)
\(29\)	−2.00000			−0.371391			−0.185695	−	0.982607i	\(-0.559454\pi\)
							−0.185695	+	0.982607i	\(0.559454\pi\)
\(31\)	−6.67026			−1.19801			−0.599007	−	0.800743i	\(-0.704438\pi\)
							−0.599007	+	0.800743i	\(0.704438\pi\)
\(37\)	2.00000			0.328798			0.164399	−	0.986394i	\(-0.447432\pi\)
							0.164399	+	0.986394i	\(0.447432\pi\)
\(41\)			7.12311i			1.11244i	0.831034	+	0.556221i	\(0.187749\pi\)
							−0.831034	+	0.556221i	\(0.812251\pi\)
\(43\)			7.60669i			1.16001i	0.814613	+	0.580005i	\(0.196949\pi\)
							−0.814613	+	0.580005i	\(0.803051\pi\)
\(47\)	10.0054			1.45944			0.729719	−	0.683748i	\(-0.239651\pi\)
							0.729719	+	0.683748i	\(0.239651\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.2.g.c.111.7	yes	8
3.2	odd	2		1260.2.c.c.811.1		8
4.3	odd	2	inner	140.2.g.c.111.6	yes	8
5.2	odd	4		700.2.c.j.699.4		8
5.3	odd	4		700.2.c.i.699.5		8
5.4	even	2		700.2.g.j.251.2		8
7.2	even	3		980.2.o.e.31.6		16
7.3	odd	6		980.2.o.e.411.1		16
7.4	even	3		980.2.o.e.411.2		16
7.5	odd	6		980.2.o.e.31.5		16
7.6	odd	2	inner	140.2.g.c.111.8	yes	8
8.3	odd	2		2240.2.k.e.1791.3		8
8.5	even	2		2240.2.k.e.1791.5		8
12.11	even	2		1260.2.c.c.811.3		8
20.3	even	4		700.2.c.i.699.3		8
20.7	even	4		700.2.c.j.699.6		8
20.19	odd	2		700.2.g.j.251.3		8
21.20	even	2		1260.2.c.c.811.2		8
28.3	even	6		980.2.o.e.411.6		16
28.11	odd	6		980.2.o.e.411.5		16
28.19	even	6		980.2.o.e.31.2		16
28.23	odd	6		980.2.o.e.31.1		16
28.27	even	2	inner	140.2.g.c.111.5	✓	8
35.13	even	4		700.2.c.j.699.5		8
35.27	even	4		700.2.c.i.699.4		8
35.34	odd	2		700.2.g.j.251.1		8
56.13	odd	2		2240.2.k.e.1791.4		8
56.27	even	2		2240.2.k.e.1791.6		8
84.83	odd	2		1260.2.c.c.811.4		8
140.27	odd	4		700.2.c.i.699.6		8
140.83	odd	4		700.2.c.j.699.3		8
140.139	even	2		700.2.g.j.251.4		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.g.c.111.5	✓	8	28.27	even	2	inner
140.2.g.c.111.6	yes	8	4.3	odd	2	inner
140.2.g.c.111.7	yes	8	1.1	even	1	trivial
140.2.g.c.111.8	yes	8	7.6	odd	2	inner
700.2.c.i.699.3		8	20.3	even	4
700.2.c.i.699.4		8	35.27	even	4
700.2.c.i.699.5		8	5.3	odd	4
700.2.c.i.699.6		8	140.27	odd	4
700.2.c.j.699.3		8	140.83	odd	4
700.2.c.j.699.4		8	5.2	odd	4
700.2.c.j.699.5		8	35.13	even	4
700.2.c.j.699.6		8	20.7	even	4
700.2.g.j.251.1		8	35.34	odd	2
700.2.g.j.251.2		8	5.4	even	2
700.2.g.j.251.3		8	20.19	odd	2
700.2.g.j.251.4		8	140.139	even	2
980.2.o.e.31.1		16	28.23	odd	6
980.2.o.e.31.2		16	28.19	even	6
980.2.o.e.31.5		16	7.5	odd	6
980.2.o.e.31.6		16	7.2	even	3
980.2.o.e.411.1		16	7.3	odd	6
980.2.o.e.411.2		16	7.4	even	3
980.2.o.e.411.5		16	28.11	odd	6
980.2.o.e.411.6		16	28.3	even	6
1260.2.c.c.811.1		8	3.2	odd	2
1260.2.c.c.811.2		8	21.20	even	2
1260.2.c.c.811.3		8	12.11	even	2
1260.2.c.c.811.4		8	84.83	odd	2
2240.2.k.e.1791.3		8	8.3	odd	2
2240.2.k.e.1791.4		8	56.13	odd	2
2240.2.k.e.1791.5		8	8.5	even	2
2240.2.k.e.1791.6		8	56.27	even	2