Embedded newform 483.2.bf.a.178.22

• Label: 483.2.bf.a.178.22
• Level: $483$
• Weight: $2$
• Character: 483.178
• 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			178.22
Character	\(\chi\)	\(=\)	483.178
Dual form			483.2.bf.a.19.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.10657 - 0.105665i) q^{2} +(-0.690079 - 0.723734i) q^{3} +(-0.750523 + 0.144651i) q^{4} +(3.37131 + 1.73803i) q^{5} +(-0.840095 - 0.727946i) q^{6} +(-0.627390 + 2.57029i) q^{7} +(-2.94838 + 0.865721i) q^{8} +(-0.0475819 + 0.998867i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.10657	−	0.105665i	0.782464	−	0.0747162i	0.303825	−	0.952728i	\(-0.401736\pi\)
							0.478639	+	0.878012i	\(0.341130\pi\)
\(3\)	−0.690079	−	0.723734i	−0.398417	−	0.417848i
							\(5\)	3.37131	+	1.73803i	1.50770	+	0.777272i	0.996611	−	0.0822577i	\(-0.0262131\pi\)
0.511086	+	0.859530i	\(0.329243\pi\)
\(7\)	−0.627390	+	2.57029i	−0.237131	+	0.971478i
							\(11\)	−0.285682	+	2.99180i	−0.0861364	+	0.902061i	0.844364	+	0.535770i	\(0.179979\pi\)
−0.930500	+	0.366291i	\(0.880627\pi\)
\(13\)	−2.29335	+	0.329734i	−0.636060	+	0.0914516i	−0.452799	−	0.891613i	\(-0.649574\pi\)
							−0.183261	+	0.983064i	\(0.558665\pi\)
\(17\)	1.19799	+	3.46136i	0.290554	+	0.839502i	0.991939	+	0.126719i	\(0.0404445\pi\)
							−0.701384	+	0.712783i	\(0.747434\pi\)
\(19\)	2.28076	−	6.58983i	0.523242	−	1.51181i	−0.305507	−	0.952190i	\(-0.598826\pi\)
							0.828749	−	0.559620i	\(-0.189053\pi\)
\(23\)	−1.59110	+	4.52420i	−0.331768	+	0.943361i
							\(29\)	6.18274	−	7.13526i	1.14811	−	1.32499i	0.210376	−	0.977620i	\(-0.432531\pi\)
0.937730	−	0.347365i	\(-0.112923\pi\)
\(31\)	2.57633	+	0.625012i	0.462723	+	0.112255i	0.460342	−	0.887742i	\(-0.347727\pi\)
							0.00238127	+	0.999997i	\(0.499242\pi\)
\(37\)	−2.24970	−	0.107166i	−0.369848	−	0.0176180i	−0.138165	−	0.990409i	\(-0.544120\pi\)
							−0.231683	+	0.972791i	\(0.574423\pi\)
\(41\)	3.73314	+	5.80888i	0.583018	+	0.907194i	0.999999	−	0.00172136i	\(-0.000547926\pi\)
							−0.416980	+	0.908916i	\(0.636912\pi\)
\(43\)	1.76468	−	6.00996i	0.269112	−	0.916511i	−0.708432	−	0.705779i	\(-0.750597\pi\)
							0.977544	−	0.210732i	\(-0.0675847\pi\)
\(47\)	−7.19644	+	4.15486i	−1.04971	+	0.606049i	−0.922568	−	0.385835i	\(-0.873913\pi\)
							−0.127140	+	0.991885i	\(0.540580\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.178.22	yes	640
7.5	odd	6	inner	483.2.bf.a.40.11	yes	640
23.19	odd	22	inner	483.2.bf.a.157.11	yes	640
161.19	even	66	inner	483.2.bf.a.19.22	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.19.22	✓	640	161.19	even	66	inner
483.2.bf.a.40.11	yes	640	7.5	odd	6	inner
483.2.bf.a.157.11	yes	640	23.19	odd	22	inner
483.2.bf.a.178.22	yes	640	1.1	even	1	trivial