Embedded newform 700.2.bj.a.323.52

• Label: 700.2.bj.a.323.52
• 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			323.52
Character	\(\chi\)	\(=\)	700.323
Dual form			700.2.bj.a.687.52

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.305820 - 1.38075i) q^{2} +(-1.10646 - 2.17156i) q^{3} +(-1.81295 - 0.844524i) q^{4} +(-2.06802 + 0.850456i) q^{5} +(-3.33676 + 0.863645i) q^{6} +(0.707107 + 0.707107i) q^{7} +(-1.72051 + 2.24496i) q^{8} +(-1.72805 + 2.37845i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 720 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.305820	−	1.38075i	0.216248	−	0.976339i
							\(3\)	−1.10646	−	2.17156i	−0.638818	−	1.25375i	−0.952591	−	0.304252i	\(-0.901593\pi\)
0.313774	−	0.949498i	\(-0.398407\pi\)
\(5\)	−2.06802	+	0.850456i	−0.924849	+	0.380335i
							\(7\)	0.707107	+	0.707107i	0.267261	+	0.267261i
\(11\)	3.23645	+	4.45459i	0.975825	+	1.34311i								0.939049	+	0.343785i	\(0.111709\pi\)
							0.0367766	+	0.999324i	\(0.488291\pi\)
\(13\)	0.0687753	+	0.434230i	0.0190748	+	0.120434i	0.995388	−	0.0959266i	\(-0.0305814\pi\)
							−0.976314	+	0.216360i	\(0.930581\pi\)
\(17\)	−5.79720	−	2.95382i	−1.40603	−	0.716407i	−0.424092	−	0.905619i	\(-0.639407\pi\)
							−0.981936	+	0.189212i	\(0.939407\pi\)
\(19\)	−0.241274	−	0.742564i	−0.0553520	−	0.170356i	0.919559	−	0.392953i	\(-0.128547\pi\)
							−0.974910	+	0.222597i	\(0.928547\pi\)
\(23\)	−1.43496	+	9.05997i	−0.299210	+	1.88914i	0.139048	+	0.990286i	\(0.455596\pi\)
							−0.438257	+	0.898849i	\(0.644404\pi\)
\(29\)	−1.88026	−	0.610934i	−0.349156	−	0.113448i	0.129188	−	0.991620i	\(-0.458763\pi\)
							−0.478344	+	0.878173i	\(0.658763\pi\)
\(31\)	1.84490	−	0.599446i	0.331354	−	0.107664i	−0.138615	−	0.990346i	\(-0.544265\pi\)
							0.469969	+	0.882683i	\(0.344265\pi\)
\(37\)	−8.66758	+	1.37281i	−1.42494	+	0.225689i	−0.820813	−	0.571198i	\(-0.806479\pi\)
							−0.604129	+	0.796886i	\(0.706479\pi\)
\(41\)	9.30492	+	6.76042i	1.45318	+	1.05580i	0.985074	+	0.172130i	\(0.0550649\pi\)
							0.468110	+	0.883670i	\(0.344935\pi\)
\(43\)	4.81086	−	4.81086i	0.733650	−	0.733650i	−0.237691	−	0.971341i	\(-0.576391\pi\)
							0.971341	+	0.237691i	\(0.0763906\pi\)
\(47\)	1.83325	−	0.934089i	0.267407	−	0.136251i	−0.315146	−	0.949043i	\(-0.602054\pi\)
							0.582554	+	0.812792i	\(0.302054\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.bj.a.323.52	yes	720
4.3	odd	2	inner	700.2.bj.a.323.49	✓	720
25.12	odd	20	inner	700.2.bj.a.687.49	yes	720
100.87	even	20	inner	700.2.bj.a.687.52	yes	720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.bj.a.323.49	✓	720	4.3	odd	2	inner
700.2.bj.a.323.52	yes	720	1.1	even	1	trivial
700.2.bj.a.687.49	yes	720	25.12	odd	20	inner
700.2.bj.a.687.52	yes	720	100.87	even	20	inner