Embedded newform 483.2.bf.a.388.22

• Label: 483.2.bf.a.388.22
• Level: $483$
• Weight: $2$
• Character: 483.388
• Analytic conductor: $3.857$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.bf (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(32\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			388.22
Character	\(\chi\)	\(=\)	483.388
Dual form			483.2.bf.a.61.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.889863 - 0.699796i) q^{2} +(-0.814576 + 0.580057i) q^{3} +(-0.169376 + 0.698177i) q^{4} +(1.17142 + 0.225774i) q^{5} +(-0.318940 + 1.08621i) q^{6} +(-2.56784 + 0.637321i) q^{7} +(1.27841 + 2.79934i) q^{8} +(0.327068 - 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.889863	−	0.699796i	0.629228	−	0.494831i	−0.251872	−	0.967761i	\(-0.581046\pi\)
							0.881100	+	0.472930i	\(0.156804\pi\)
\(3\)	−0.814576	+	0.580057i	−0.470296	+	0.334896i
							\(5\)	1.17142	+	0.225774i	0.523877	+	0.100969i	0.444332	−	0.895862i	\(-0.353441\pi\)
0.0795451	+	0.996831i	\(0.474653\pi\)
\(7\)	−2.56784	+	0.637321i	−0.970554	+	0.240885i
							\(11\)	−1.31250	+	1.66898i	−0.395734	+	0.503217i	−0.942563	−	0.334027i	\(-0.891592\pi\)
0.546830	+	0.837244i	\(0.315835\pi\)
\(13\)	−0.833274	+	1.29660i	−0.231109	+	0.359612i	−0.937368	−	0.348341i	\(-0.886745\pi\)
							0.706259	+	0.707954i	\(0.250381\pi\)
\(17\)	−1.55405	+	1.48178i	−0.376912	+	0.359385i	−0.854729	−	0.519075i	\(-0.826277\pi\)
							0.477817	+	0.878459i	\(0.341428\pi\)
\(19\)	1.67423	+	1.59637i	0.384095	+	0.366234i	0.857387	−	0.514672i	\(-0.172086\pi\)
							−0.473293	+	0.880905i	\(0.656935\pi\)
\(23\)	−1.40646	+	4.58496i	−0.293267	+	0.956030i
							\(29\)	8.02958	+	2.35770i	1.49106	+	0.437813i	0.922878	−	0.385093i	\(-0.125831\pi\)
0.568178	+	0.822906i	\(0.307649\pi\)
\(31\)	0.0120150	−	0.125827i	0.00215796	−	0.0225992i	−0.994329	−	0.106350i	\(-0.966084\pi\)
							0.996487	+	0.0837503i	\(0.0266898\pi\)
\(37\)	7.68954	+	2.66138i	1.26415	+	0.437527i	0.875214	−	0.483736i	\(-0.160721\pi\)
							0.388939	+	0.921264i	\(0.372842\pi\)
\(41\)	−8.08770	−	7.00803i	−1.26309	−	1.09447i	−0.991232	−	0.132133i	\(-0.957818\pi\)
							−0.271855	−	0.962338i	\(-0.587637\pi\)
\(43\)	6.11155	+	2.79105i	0.932003	+	0.425632i	0.822764	−	0.568384i	\(-0.192431\pi\)
							0.109240	+	0.994015i	\(0.465158\pi\)
\(47\)	2.18729	−	1.26283i	0.319049	−	0.184203i	−0.331919	−	0.943308i	\(-0.607696\pi\)
							0.650969	+	0.759105i	\(0.274363\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.388.22	yes	640
7.5	odd	6	inner	483.2.bf.a.250.11	yes	640
23.15	odd	22	inner	483.2.bf.a.199.11	yes	640
161.61	even	66	inner	483.2.bf.a.61.22	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.22	✓	640	161.61	even	66	inner
483.2.bf.a.199.11	yes	640	23.15	odd	22	inner
483.2.bf.a.250.11	yes	640	7.5	odd	6	inner
483.2.bf.a.388.22	yes	640	1.1	even	1	trivial