Embedded newform 700.2.p.c.451.11

• Label: 700.2.p.c.451.11
• Level: $700$
• Weight: $2$
• Character: 700.451
• Analytic conductor: $5.590$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	700.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.58952814149\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			451.11
Character	\(\chi\)	\(=\)	700.451
Dual form			700.2.p.c.551.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.501653 + 1.32225i) q^{2} +(0.895374 - 1.55083i) q^{3} +(-1.49669 + 1.32662i) q^{4} +(2.49976 + 0.405928i) q^{6} +(0.644798 - 2.56598i) q^{7} +(-2.50494 - 1.31349i) q^{8} +(-0.103389 - 0.179074i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{2} - 2 q^{4} + 4 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(351\)	\(477\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.501653	+	1.32225i	0.354722	+	0.934972i
							\(3\)	0.895374	−	1.55083i	0.516944	−	0.895374i	−0.482862	−	0.875696i	\(-0.660403\pi\)
0.999806	−	0.0196774i	\(-0.00626391\pi\)
\(5\)		0			0
							\(7\)	0.644798	−	2.56598i	0.243711	−	0.969848i
\(11\)	3.66757	+	2.11747i	1.10581	+	0.638442i								0.937742	−	0.347333i	\(-0.112913\pi\)
							0.168072	+	0.985775i	\(0.446246\pi\)
\(13\)		−	2.98261i		−	0.827227i	−0.910453	−	0.413613i	\(-0.864267\pi\)
							0.910453	−	0.413613i	\(-0.135733\pi\)
\(17\)	1.92087	+	1.10901i	0.465879	+	0.268976i	0.714513	−	0.699622i	\(-0.246648\pi\)
							−0.248634	+	0.968598i	\(0.579982\pi\)
\(19\)	−2.28341	−	3.95499i	−0.523851	−	0.907336i	−0.999615	−	0.0277631i	\(-0.991162\pi\)
							0.475764	−	0.879573i	\(-0.342172\pi\)
\(23\)	1.78388	−	1.02992i	0.371965	−	0.214754i	−0.302352	−	0.953196i	\(-0.597772\pi\)
							0.674316	+	0.738443i	\(0.264438\pi\)
\(29\)	6.42784			1.19362			0.596810	−	0.802383i	\(-0.296435\pi\)
							0.596810	+	0.802383i	\(0.296435\pi\)
\(31\)	1.20072	−	2.07971i	0.215656	−	0.373527i	−0.737819	−	0.674998i	\(-0.764144\pi\)
							0.953475	+	0.301471i	\(0.0974777\pi\)
\(37\)	−2.16224	−	3.74511i	−0.355470	−	0.615692i	0.631728	−	0.775190i	\(-0.282346\pi\)
							−0.987198	+	0.159498i	\(0.949013\pi\)
\(41\)			4.88552i			0.762990i	0.924371	+	0.381495i	\(0.124591\pi\)
							−0.924371	+	0.381495i	\(0.875409\pi\)
\(43\)			12.3733i			1.88692i	0.331490	+	0.943459i	\(0.392449\pi\)
							−0.331490	+	0.943459i	\(0.607551\pi\)
\(47\)	−3.38231	−	5.85833i	−0.493360	−	0.854525i	0.506610	−	0.862175i	\(-0.330898\pi\)
							−0.999971	+	0.00764982i	\(0.997565\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.p.c.451.11		32
4.3	odd	2	inner	700.2.p.c.451.12		32
5.2	odd	4		700.2.t.c.199.2		32
5.3	odd	4		700.2.t.d.199.15		32
5.4	even	2		140.2.o.a.31.6	yes	32
7.5	odd	6	inner	700.2.p.c.551.12		32
20.3	even	4		700.2.t.d.199.14		32
20.7	even	4		700.2.t.c.199.3		32
20.19	odd	2		140.2.o.a.31.5	✓	32
28.19	even	6	inner	700.2.p.c.551.11		32
35.4	even	6		980.2.g.a.391.32		32
35.9	even	6		980.2.o.f.411.5		32
35.12	even	12		700.2.t.d.299.14		32
35.19	odd	6		140.2.o.a.131.5	yes	32
35.24	odd	6		980.2.g.a.391.31		32
35.33	even	12		700.2.t.c.299.3		32
35.34	odd	2		980.2.o.f.31.6		32
140.19	even	6		140.2.o.a.131.6	yes	32
140.39	odd	6		980.2.g.a.391.29		32
140.47	odd	12		700.2.t.d.299.15		32
140.59	even	6		980.2.g.a.391.30		32
140.79	odd	6		980.2.o.f.411.6		32
140.103	odd	12		700.2.t.c.299.2		32
140.139	even	2		980.2.o.f.31.5		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.o.a.31.5	✓	32	20.19	odd	2
140.2.o.a.31.6	yes	32	5.4	even	2
140.2.o.a.131.5	yes	32	35.19	odd	6
140.2.o.a.131.6	yes	32	140.19	even	6
700.2.p.c.451.11		32	1.1	even	1	trivial
700.2.p.c.451.12		32	4.3	odd	2	inner
700.2.p.c.551.11		32	28.19	even	6	inner
700.2.p.c.551.12		32	7.5	odd	6	inner
700.2.t.c.199.2		32	5.2	odd	4
700.2.t.c.199.3		32	20.7	even	4
700.2.t.c.299.2		32	140.103	odd	12
700.2.t.c.299.3		32	35.33	even	12
700.2.t.d.199.14		32	20.3	even	4
700.2.t.d.199.15		32	5.3	odd	4
700.2.t.d.299.14		32	35.12	even	12
700.2.t.d.299.15		32	140.47	odd	12
980.2.g.a.391.29		32	140.39	odd	6
980.2.g.a.391.30		32	140.59	even	6
980.2.g.a.391.31		32	35.24	odd	6
980.2.g.a.391.32		32	35.4	even	6
980.2.o.f.31.5		32	140.139	even	2
980.2.o.f.31.6		32	35.34	odd	2
980.2.o.f.411.5		32	35.9	even	6
980.2.o.f.411.6		32	140.79	odd	6