Embedded newform 483.2.bf.a.19.17

• Label: 483.2.bf.a.19.17
• Level: $483$
• Weight: $2$
• Character: 483.19
• 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			19.17
Character	\(\chi\)	\(=\)	483.19
Dual form			483.2.bf.a.178.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.174212 + 0.0166352i) q^{2} +(-0.690079 + 0.723734i) q^{3} +(-1.93378 - 0.372706i) q^{4} +(0.315538 - 0.162671i) q^{5} +(-0.132259 + 0.114603i) q^{6} +(2.63128 + 0.276333i) q^{7} +(-0.666518 - 0.195707i) q^{8} +(-0.0475819 - 0.998867i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{15}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.174212	+	0.0166352i	0.123186	+	0.0117629i	0.156467	−	0.987683i	\(-0.449990\pi\)
							−0.0332808	+	0.999446i	\(0.510596\pi\)
\(3\)	−0.690079	+	0.723734i	−0.398417	+	0.417848i
							\(5\)	0.315538	−	0.162671i	0.141113	−	0.0727487i	−0.386224	−	0.922405i	\(-0.626221\pi\)
0.527337	+	0.849656i	\(0.323191\pi\)
\(7\)	2.63128	+	0.276333i	0.994531	+	0.104444i
							\(11\)	0.132286	+	1.38536i	0.0398857	+	0.417702i	0.993260	+	0.115910i	\(0.0369786\pi\)
−0.953374	+	0.301791i	\(0.902415\pi\)
\(13\)	−0.399449	−	0.0574321i	−0.110787	−	0.0159288i	0.0866982	−	0.996235i	\(-0.472368\pi\)
							−0.197485	+	0.980306i	\(0.563278\pi\)
\(17\)	0.00865619	−	0.0250104i	0.00209943	−	0.00606592i	−0.943946	−	0.330099i	\(-0.892918\pi\)
							0.946046	+	0.324033i	\(0.105039\pi\)
\(19\)	1.86799	+	5.39719i	0.428545	+	1.23820i	0.928235	+	0.371995i	\(0.121326\pi\)
							−0.499689	+	0.866205i	\(0.666553\pi\)
\(23\)	4.33054	+	2.06067i	0.902981	+	0.429680i
							\(29\)	3.18409	+	3.67464i	0.591271	+	0.682363i	0.969989	−	0.243149i	\(-0.0781805\pi\)
−0.378718	+	0.925512i	\(0.623635\pi\)
\(31\)	4.35868	−	1.05741i	0.782843	−	0.189916i	0.175631	−	0.984456i	\(-0.443803\pi\)
							0.607211	+	0.794540i	\(0.292288\pi\)
\(37\)	8.03721	−	0.382860i	1.32131	−	0.0629417i	0.624967	−	0.780651i	\(-0.285112\pi\)
							0.696343	+	0.717710i	\(0.254809\pi\)
\(41\)	−6.16670	+	9.59557i	−0.963077	+	1.49858i	−0.0990797	+	0.995079i	\(0.531590\pi\)
							−0.863997	+	0.503497i	\(0.832046\pi\)
\(43\)	−0.194037	−	0.660829i	−0.0295903	−	0.100775i	0.943374	−	0.331732i	\(-0.107633\pi\)
							−0.972964	+	0.230956i	\(0.925815\pi\)
\(47\)	−0.862571	−	0.498005i	−0.125819	−	0.0726415i	0.435770	−	0.900058i	\(-0.356476\pi\)
							−0.561588	+	0.827417i	\(0.689810\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.19.17	✓	640
7.3	odd	6	inner	483.2.bf.a.157.16	yes	640
23.17	odd	22	inner	483.2.bf.a.40.16	yes	640
161.17	even	66	inner	483.2.bf.a.178.17	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.19.17	✓	640	1.1	even	1	trivial
483.2.bf.a.40.16	yes	640	23.17	odd	22	inner
483.2.bf.a.157.16	yes	640	7.3	odd	6	inner
483.2.bf.a.178.17	yes	640	161.17	even	66	inner