Embedded newform 700.2.bj.a.323.12

• Label: 700.2.bj.a.323.12
• 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			323.12
Character	\(\chi\)	\(=\)	700.323
Dual form			700.2.bj.a.687.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.32798 + 0.486287i) q^{2} +(-1.16698 - 2.29033i) q^{3} +(1.52705 - 1.29156i) q^{4} +(-1.22764 - 1.86893i) q^{5} +(2.66349 + 2.47402i) q^{6} +(0.707107 + 0.707107i) q^{7} +(-1.39982 + 2.45774i) q^{8} +(-2.12042 + 2.91850i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 720 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32798	+	0.486287i	−0.939022	+	0.343857i
							\(3\)	−1.16698	−	2.29033i	−0.673758	−	1.32232i	−0.934171	−	0.356827i	\(-0.883859\pi\)
0.260413	−	0.965497i	\(-0.416141\pi\)
\(5\)	−1.22764	−	1.86893i	−0.549019	−	0.835810i
							\(7\)	0.707107	+	0.707107i	0.267261	+	0.267261i
\(11\)	−1.52142	−	2.09405i	−0.458725	−	0.631381i								0.515519	−	0.856878i	\(-0.327599\pi\)
							−0.974244	+	0.225498i	\(0.927599\pi\)
\(13\)	1.01891	+	6.43312i	0.282594	+	1.78423i	0.565161	+	0.824980i	\(0.308814\pi\)
							−0.282568	+	0.959247i	\(0.591186\pi\)
\(17\)	−6.85573	−	3.49317i	−1.66276	−	0.847218i	−0.994667	−	0.103140i	\(-0.967111\pi\)
							−0.668092	−	0.744078i	\(-0.732889\pi\)
\(19\)	−1.13155	−	3.48255i	−0.259595	−	0.798951i	−0.992889	−	0.119040i	\(-0.962018\pi\)
							0.733295	−	0.679911i	\(-0.237982\pi\)
\(23\)	−0.788484	+	4.97829i	−0.164410	+	1.03805i	0.758118	+	0.652117i	\(0.226119\pi\)
							−0.922528	+	0.385929i	\(0.873881\pi\)
\(29\)	3.64389	+	1.18397i	0.676653	+	0.219858i	0.627129	−	0.778915i	\(-0.284230\pi\)
							0.0495235	+	0.998773i	\(0.484230\pi\)
\(31\)	−3.56893	+	1.15962i	−0.640999	+	0.208273i	−0.611441	−	0.791290i	\(-0.709410\pi\)
							−0.0295576	+	0.999563i	\(0.509410\pi\)
\(37\)	8.69766	−	1.37757i	1.42989	−	0.226472i	0.607011	−	0.794693i	\(-0.292368\pi\)
							0.822875	+	0.568222i	\(0.192368\pi\)
\(41\)	1.06920	+	0.776820i	0.166981	+	0.121319i	0.668137	−	0.744038i	\(-0.267092\pi\)
							−0.501156	+	0.865357i	\(0.667092\pi\)
\(43\)	−3.97309	+	3.97309i	−0.605891	+	0.605891i	−0.941870	−	0.335978i	\(-0.890933\pi\)
							0.335978	+	0.941870i	\(0.390933\pi\)
\(47\)	−6.64051	+	3.38351i	−0.968617	+	0.493535i	−0.865377	−	0.501121i	\(-0.832921\pi\)
							−0.103240	+	0.994656i	\(0.532921\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.bj.a.323.12	✓	720
4.3	odd	2	inner	700.2.bj.a.323.71	yes	720
25.12	odd	20	inner	700.2.bj.a.687.71	yes	720
100.87	even	20	inner	700.2.bj.a.687.12	yes	720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.bj.a.323.12	✓	720	1.1	even	1	trivial
700.2.bj.a.323.71	yes	720	4.3	odd	2	inner
700.2.bj.a.687.12	yes	720	100.87	even	20	inner
700.2.bj.a.687.71	yes	720	25.12	odd	20	inner