Embedded newform 483.2.bf.a.178.25

• Label: 483.2.bf.a.178.25
• 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			178.25
Character	\(\chi\)	\(=\)	483.178
Dual form			483.2.bf.a.19.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.63043 - 0.155687i) q^{2} +(0.690079 + 0.723734i) q^{3} +(0.670191 - 0.129169i) q^{4} +(3.55311 + 1.83175i) q^{5} +(1.23780 + 1.07256i) q^{6} +(-2.60509 - 0.462092i) q^{7} +(-2.07041 + 0.607927i) q^{8} +(-0.0475819 + 0.998867i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.63043	−	0.155687i	1.15288	−	0.110087i	0.498945	−	0.866634i	\(-0.333721\pi\)
							0.653940	+	0.756546i	\(0.273115\pi\)
\(3\)	0.690079	+	0.723734i	0.398417	+	0.417848i
							\(5\)	3.55311	+	1.83175i	1.58900	+	0.819185i	0.999938	+	0.0110947i	\(0.00353164\pi\)
0.589059	+	0.808090i	\(0.299499\pi\)
\(7\)	−2.60509	−	0.462092i	−0.984630	−	0.174654i
							\(11\)	0.0284877	−	0.298336i	0.00858935	−	0.0899518i	−0.990161	−	0.139935i	\(-0.955311\pi\)
0.998750	+	0.0499831i	\(0.0159167\pi\)
\(13\)	5.98866	−	0.861039i	1.66096	−	0.238809i	0.753044	−	0.657971i	\(-0.228585\pi\)
							0.907912	+	0.419161i	\(0.137676\pi\)
\(17\)	−0.680955	−	1.96749i	−0.165156	−	0.477187i	0.831807	−	0.555065i	\(-0.187307\pi\)
							−0.996963	+	0.0778783i	\(0.975185\pi\)
\(19\)	−0.499556	+	1.44337i	−0.114606	+	0.331132i	−0.987687	−	0.156443i	\(-0.949997\pi\)
							0.873081	+	0.487575i	\(0.162119\pi\)
\(23\)	−3.35233	−	3.42956i	−0.699010	−	0.715112i
							\(29\)	1.92778	−	2.22478i	0.357980	−	0.413131i	−0.547982	−	0.836490i	\(-0.684604\pi\)
0.905962	+	0.423359i	\(0.139149\pi\)
\(31\)	−6.99100	−	1.69600i	−1.25562	−	0.304610i	−0.447850	−	0.894108i	\(-0.647810\pi\)
							−0.807770	+	0.589498i	\(0.799325\pi\)
\(37\)	1.65123	+	0.0786579i	0.271461	+	0.0129313i	0.182871	−	0.983137i	\(-0.441461\pi\)
							0.0885899	+	0.996068i	\(0.471764\pi\)
\(41\)	−3.47422	−	5.40599i	−0.542582	−	0.844273i	0.456382	−	0.889784i	\(-0.349145\pi\)
							−0.998964	+	0.0455104i	\(0.985509\pi\)
\(43\)	3.04969	−	10.3863i	0.465073	−	1.58389i	−0.309172	−	0.951006i	\(-0.600052\pi\)
							0.774245	−	0.632886i	\(-0.218130\pi\)
\(47\)	7.29260	−	4.21038i	1.06373	−	0.614148i	0.137272	−	0.990533i	\(-0.456167\pi\)
							0.926463	+	0.376386i	\(0.122833\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.178.25	yes	640
7.5	odd	6	inner	483.2.bf.a.40.8	yes	640
23.19	odd	22	inner	483.2.bf.a.157.8	yes	640
161.19	even	66	inner	483.2.bf.a.19.25	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.19.25	✓	640	161.19	even	66	inner
483.2.bf.a.40.8	yes	640	7.5	odd	6	inner
483.2.bf.a.157.8	yes	640	23.19	odd	22	inner
483.2.bf.a.178.25	yes	640	1.1	even	1	trivial