Embedded newform 483.2.y.b.4.9

• Label: 483.2.y.b.4.9
• Level: $483$
• Weight: $2$
• Character: 483.4
• 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			4.9
Character	\(\chi\)	\(=\)	483.4
Dual form			483.2.y.b.121.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0919807 + 0.129169i) q^{2} +(0.235759 - 0.971812i) q^{3} +(0.645912 + 1.86624i) q^{4} +(0.0626824 - 1.31586i) q^{5} +(0.103842 + 0.119841i) q^{6} +(0.568266 - 2.58400i) q^{7} +(-0.604769 - 0.177576i) q^{8} +(-0.888835 - 0.458227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0919807	+	0.129169i	−0.0650402	+	0.0913361i	−0.845815	−	0.533477i	\(-0.820885\pi\)
							0.780775	+	0.624813i	\(0.214825\pi\)
\(3\)	0.235759	−	0.971812i	0.136115	−	0.561076i
							\(5\)	0.0626824	−	1.31586i	0.0280324	−	0.588473i	−0.940531	−	0.339707i	\(-0.889672\pi\)
0.968564	−	0.248766i	\(-0.0800249\pi\)
\(7\)	0.568266	−	2.58400i	0.214785	−	0.976661i
							\(11\)	−1.86588	−	2.62026i	−0.562583	−	0.790038i	0.431094	−	0.902307i	\(-0.358128\pi\)
−0.993678	+	0.112269i	\(0.964188\pi\)
\(13\)	0.717504	−	4.99035i	0.199000	−	1.38407i	−0.608195	−	0.793788i	\(-0.708106\pi\)
							0.807195	−	0.590286i	\(-0.200985\pi\)
\(17\)	−2.10666	+	0.406024i	−0.510939	+	0.0984754i	−0.438203	−	0.898876i	\(-0.644385\pi\)
							−0.0727355	+	0.997351i	\(0.523173\pi\)
\(19\)	5.62935	+	1.08497i	1.29146	+	0.248909i	0.788304	−	0.615286i	\(-0.210959\pi\)
							0.503158	+	0.864195i	\(0.332171\pi\)
\(23\)	3.12217	+	3.64034i	0.651017	+	0.759063i
							\(29\)	−6.22697	−	7.18631i	−1.15632	−	1.33446i	−0.933066	−	0.359704i	\(-0.882878\pi\)
−0.223254	−	0.974760i	\(-0.571668\pi\)
\(31\)	2.58605	−	2.46580i	0.464469	−	0.442870i	−0.421352	−	0.906897i	\(-0.638444\pi\)
							0.885821	+	0.464027i	\(0.153596\pi\)
\(37\)	3.24395	+	1.67237i	0.533302	+	0.274936i	0.703771	−	0.710427i	\(-0.251498\pi\)
							−0.170469	+	0.985363i	\(0.554528\pi\)
\(41\)	7.80449	+	5.01564i	1.21886	+	0.783312i	0.982119	−	0.188261i	\(-0.0602852\pi\)
							0.236738	+	0.971573i	\(0.423922\pi\)
\(43\)	−6.63141	+	1.94716i	−1.01128	+	0.296939i	−0.745077	−	0.666978i	\(-0.767587\pi\)
							−0.266203	+	0.963917i	\(0.585769\pi\)
\(47\)	−0.750846	−	1.30050i	−0.109522	−	0.189698i	0.806055	−	0.591841i	\(-0.201599\pi\)
							−0.915577	+	0.402143i	\(0.868265\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.4.9	✓	320
7.2	even	3	inner	483.2.y.b.142.8	yes	320
23.6	even	11	inner	483.2.y.b.466.8	yes	320
161.121	even	33	inner	483.2.y.b.121.9	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.4.9	✓	320	1.1	even	1	trivial
483.2.y.b.121.9	yes	320	161.121	even	33	inner
483.2.y.b.142.8	yes	320	7.2	even	3	inner
483.2.y.b.466.8	yes	320	23.6	even	11	inner