Embedded newform 483.2.bf.a.178.1

• Label: 483.2.bf.a.178.1
• 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			178.1
Character	\(\chi\)	\(=\)	483.178
Dual form			483.2.bf.a.19.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.78242 + 0.265689i) q^{2} +(0.690079 + 0.723734i) q^{3} +(5.70742 - 1.10001i) q^{4} +(0.0901076 + 0.0464537i) q^{5} +(-2.11238 - 1.83039i) q^{6} +(2.63520 + 0.236108i) q^{7} +(-10.2245 + 3.00217i) q^{8} +(-0.0475819 + 0.998867i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.78242	+	0.265689i	−1.96747	+	0.187870i	−0.998288	−	0.0584833i	\(-0.981374\pi\)
							−0.969180	+	0.246354i	\(0.920767\pi\)
\(3\)	0.690079	+	0.723734i	0.398417	+	0.417848i
							\(5\)	0.0901076	+	0.0464537i	0.0402973	+	0.0207747i	0.478257	−	0.878220i	\(-0.341269\pi\)
−0.437960	+	0.898994i	\(0.644299\pi\)
\(7\)	2.63520	+	0.236108i	0.996010	+	0.0892403i
							\(11\)	−0.291164	+	3.04921i	−0.0877893	+	0.919371i	0.839085	+	0.544000i	\(0.183091\pi\)
−0.926875	+	0.375371i	\(0.877515\pi\)
\(13\)	−5.20984	+	0.749062i	−1.44495	+	0.207753i	−0.819754	−	0.572715i	\(-0.805890\pi\)
							−0.625196	+	0.780468i	\(0.714981\pi\)
\(17\)	0.524494	+	1.51543i	0.127208	+	0.367545i	0.990564	−	0.137052i	\(-0.0437626\pi\)
							−0.863355	+	0.504596i	\(0.831641\pi\)
\(19\)	−2.69452	+	7.78530i	−0.618165	+	1.78607i	−2.92877e−5		1.00000i	\(0.500009\pi\)
							−0.618136	+	0.786071i	\(0.712112\pi\)
\(23\)	4.54332	−	1.53566i	0.947348	−	0.320206i
							\(29\)	2.32069	−	2.67822i	0.430942	−	0.497334i	−0.498197	−	0.867064i	\(-0.666004\pi\)
0.929139	+	0.369730i	\(0.120550\pi\)
\(31\)	4.80214	+	1.16499i	0.862490	+	0.209238i	0.642520	−	0.766269i	\(-0.277889\pi\)
							0.219970	+	0.975507i	\(0.429404\pi\)
\(37\)	−2.77761	−	0.132314i	−0.456637	−	0.0217523i	−0.181996	−	0.983299i	\(-0.558256\pi\)
							−0.274641	+	0.961547i	\(0.588559\pi\)
\(41\)	1.21272	+	1.88703i	0.189395	+	0.294705i	0.922946	−	0.384930i	\(-0.125775\pi\)
							−0.733550	+	0.679635i	\(0.762138\pi\)
\(43\)	−2.33410	+	7.94921i	−0.355947	+	1.21224i	0.565830	+	0.824522i	\(0.308556\pi\)
							−0.921777	+	0.387721i	\(0.873262\pi\)
\(47\)	1.46454	−	0.845553i	0.213625	−	0.123337i	−0.389370	−	0.921081i	\(-0.627307\pi\)
							0.602995	+	0.797745i	\(0.293974\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.178.1	yes	640
7.5	odd	6	inner	483.2.bf.a.40.32	yes	640
23.19	odd	22	inner	483.2.bf.a.157.32	yes	640
161.19	even	66	inner	483.2.bf.a.19.1	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.19.1	✓	640	161.19	even	66	inner
483.2.bf.a.40.32	yes	640	7.5	odd	6	inner
483.2.bf.a.157.32	yes	640	23.19	odd	22	inner
483.2.bf.a.178.1	yes	640	1.1	even	1	trivial