Embedded newform 700.2.bj.a.323.32

• Label: 700.2.bj.a.323.32
• Level: $700$
• Weight: $2$
• Character: 700.323
• Analytic conductor: $5.590$
• Analytic rank: $0$
• Dimension: $720$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	700.bj (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.58952814149\)
Analytic rank:	\(0\)
Dimension:	\(720\)
Relative dimension:	\(90\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			323.32
Character	\(\chi\)	\(=\)	700.323
Dual form			700.2.bj.a.687.32

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.521152 - 1.31469i) q^{2} +(1.38963 + 2.72730i) q^{3} +(-1.45680 + 1.37030i) q^{4} +(-0.550371 - 2.16728i) q^{5} +(2.86134 - 3.24827i) q^{6} +(-0.707107 - 0.707107i) q^{7} +(2.56073 + 1.20110i) q^{8} +(-3.74375 + 5.15283i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 720 q+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(351\)	\(477\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{11}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.521152	−	1.31469i	−0.368510	−	0.929624i
							\(3\)	1.38963	+	2.72730i	0.802303	+	1.57461i	0.818338	+	0.574737i	\(0.194896\pi\)
−0.0160352	+	0.999871i	\(0.505104\pi\)
\(5\)	−0.550371	−	2.16728i	−0.246134	−	0.969236i
							\(7\)	−0.707107	−	0.707107i	−0.267261	−	0.267261i
\(11\)	0.474640	+	0.653285i	0.143109	+	0.196973i								0.874555	−	0.484927i	\(-0.161154\pi\)
							−0.731445	+	0.681900i	\(0.761154\pi\)
\(13\)	−0.0801358	−	0.505957i	−0.0222257	−	0.140327i	0.974080	−	0.226203i	\(-0.0726312\pi\)
							−0.996306	+	0.0858756i	\(0.972631\pi\)
\(17\)	6.88020	+	3.50564i	1.66869	+	0.850242i	0.993656	+	0.112458i	\(0.0358725\pi\)
							0.675037	+	0.737784i	\(0.264128\pi\)
\(19\)	2.40162	+	7.39142i	0.550969	+	1.69571i	0.706359	+	0.707854i	\(0.250337\pi\)
							−0.155390	+	0.987853i	\(0.549663\pi\)
\(23\)	−0.516845	+	3.26323i	−0.107770	+	0.680431i	0.873359	+	0.487076i	\(0.161937\pi\)
							−0.981129	+	0.193354i	\(0.938063\pi\)
\(29\)	3.83855	+	1.24722i	0.712801	+	0.231603i	0.642899	−	0.765951i	\(-0.277731\pi\)
							0.0699017	+	0.997554i	\(0.477731\pi\)
\(31\)	−2.71641	+	0.882616i	−0.487882	+	0.158523i	−0.542620	−	0.839978i	\(-0.682568\pi\)
							0.0547380	+	0.998501i	\(0.482568\pi\)
\(37\)	−0.458943	+	0.0726894i	−0.0754497	+	0.0119501i	−0.194045	−	0.980993i	\(-0.562161\pi\)
							0.118595	+	0.992943i	\(0.462161\pi\)
\(41\)	−1.69993	−	1.23507i	−0.265485	−	0.192886i	0.447077	−	0.894496i	\(-0.352465\pi\)
							−0.712562	+	0.701609i	\(0.752465\pi\)
\(43\)	−3.73632	+	3.73632i	−0.569783	+	0.569783i	−0.932068	−	0.362285i	\(-0.881997\pi\)
							0.362285	+	0.932068i	\(0.381997\pi\)
\(47\)	−0.121322	+	0.0618164i	−0.0176966	+	0.00901685i	−0.462816	−	0.886454i	\(-0.653161\pi\)
							0.445120	+	0.895471i	\(0.353161\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.bj.a.323.32	✓	720
4.3	odd	2	inner	700.2.bj.a.323.66	yes	720
25.12	odd	20	inner	700.2.bj.a.687.66	yes	720
100.87	even	20	inner	700.2.bj.a.687.32	yes	720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.bj.a.323.32	✓	720	1.1	even	1	trivial
700.2.bj.a.323.66	yes	720	4.3	odd	2	inner
700.2.bj.a.687.32	yes	720	100.87	even	20	inner
700.2.bj.a.687.66	yes	720	25.12	odd	20	inner