Embedded newform 483.2.y.b.340.4

• Label: 483.2.y.b.340.4
• Level: $483$
• Weight: $2$
• Character: 483.340
• Analytic conductor: $3.857$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.y (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(16\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			340.4
Character	\(\chi\)	\(=\)	483.340
Dual form			483.2.y.b.331.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.460427 - 1.89791i) q^{2} +(-0.327068 - 0.945001i) q^{3} +(-1.61239 + 0.831245i) q^{4} +(0.128307 - 0.0513663i) q^{5} +(-1.64293 + 1.05585i) q^{6} +(-1.62090 - 2.09109i) q^{7} +(-0.237817 - 0.274456i) q^{8} +(-0.786053 + 0.618159i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{6}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.460427	−	1.89791i	−0.325571	−	1.34202i	−0.865752	−	0.500474i	\(-0.833159\pi\)
							0.540180	−	0.841549i	\(-0.318356\pi\)
\(3\)	−0.327068	−	0.945001i	−0.188833	−	0.545596i
							\(5\)	0.128307	−	0.0513663i	0.0573806	−	0.0229717i	−0.342795	−	0.939410i	\(-0.611374\pi\)
0.400175	+	0.916439i	\(0.368949\pi\)
\(7\)	−1.62090	−	2.09109i	−0.612644	−	0.790359i
							\(11\)	0.270110	−	1.11341i	0.0814412	−	0.335705i	−0.916485	−	0.400068i	\(-0.868986\pi\)
0.997927	+	0.0643630i	\(0.0205016\pi\)
\(13\)	0.526616	−	1.15313i	0.146057	−	0.319820i	−0.822437	−	0.568856i	\(-0.807386\pi\)
							0.968494	+	0.249036i	\(0.0801136\pi\)
\(17\)	−0.146768	−	3.08103i	−0.0355964	−	0.747261i	−0.944155	−	0.329503i	\(-0.893119\pi\)
							0.908558	−	0.417758i	\(-0.137184\pi\)
\(19\)	−0.226514	+	4.75511i	−0.0519658	+	1.09090i	0.810810	+	0.585310i	\(0.199027\pi\)
							−0.862776	+	0.505587i	\(0.831276\pi\)
\(23\)	−4.72631	−	0.813647i	−0.985503	−	0.169657i
							\(29\)	4.44994	−	2.85981i	0.826334	−	0.531052i	−0.0577772	−	0.998330i	\(-0.518401\pi\)
0.884111	+	0.467277i	\(0.154765\pi\)
\(31\)	−1.86125	+	0.358727i	−0.334291	+	0.0644293i	−0.353635	−	0.935384i	\(-0.615054\pi\)
							0.0193439	+	0.999813i	\(0.493842\pi\)
\(37\)	−4.39296	+	3.45466i	−0.722198	+	0.567943i	−0.910172	−	0.414231i	\(-0.864051\pi\)
							0.187973	+	0.982174i	\(0.439808\pi\)
\(41\)	0.573363	−	3.98783i	0.0895442	−	0.622794i	−0.894791	−	0.446486i	\(-0.852675\pi\)
							0.984335	−	0.176308i	\(-0.0564155\pi\)
\(43\)	4.23547	−	4.88799i	0.645903	−	0.745412i	−0.334504	−	0.942394i	\(-0.608569\pi\)
							0.980407	+	0.196983i	\(0.0631143\pi\)
\(47\)	−1.76906	−	3.06410i	−0.258043	−	0.446944i	0.707674	−	0.706539i	\(-0.249744\pi\)
							−0.965718	+	0.259594i	\(0.916411\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.340.4	yes	320
7.2	even	3	inner	483.2.y.b.478.4	yes	320
23.9	even	11	inner	483.2.y.b.193.4	✓	320
161.9	even	33	inner	483.2.y.b.331.4	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.193.4	✓	320	23.9	even	11	inner
483.2.y.b.331.4	yes	320	161.9	even	33	inner
483.2.y.b.340.4	yes	320	1.1	even	1	trivial
483.2.y.b.478.4	yes	320	7.2	even	3	inner