Embedded newform 483.2.y.b.4.3

• Label: 483.2.y.b.4.3
• 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			4.3
Character	\(\chi\)	\(=\)	483.4
Dual form			483.2.y.b.121.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.32550 + 1.86141i) q^{2} +(0.235759 - 0.971812i) q^{3} +(-1.05374 - 3.04459i) q^{4} +(-0.0837421 + 1.75796i) q^{5} +(1.49644 + 1.72698i) q^{6} +(0.00745284 - 2.64574i) q^{7} +(2.67883 + 0.786576i) q^{8} +(-0.888835 - 0.458227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32550	+	1.86141i	−0.937271	+	1.31621i	0.0113960	+	0.999935i	\(0.496372\pi\)
							−0.948667	+	0.316278i	\(0.897567\pi\)
\(3\)	0.235759	−	0.971812i	0.136115	−	0.561076i
							\(5\)	−0.0837421	+	1.75796i	−0.0374506	+	0.786185i	0.899542	+	0.436835i	\(0.143901\pi\)
−0.936992	+	0.349350i	\(0.886402\pi\)
\(7\)	0.00745284	−	2.64574i	0.00281691	−	0.999996i
							\(11\)	0.920488	+	1.29265i	0.277538	+	0.389747i	0.929691	−	0.368340i	\(-0.120074\pi\)
−0.652154	+	0.758087i	\(0.726134\pi\)
\(13\)	−0.0452576	+	0.314774i	−0.0125522	+	0.0873025i	−0.995135	−	0.0985251i	\(-0.968588\pi\)
							0.982582	+	0.185828i	\(0.0594966\pi\)
\(17\)	4.22965	−	0.815199i	1.02584	−	0.197715i	0.351542	−	0.936172i	\(-0.385658\pi\)
							0.674300	+	0.738457i	\(0.264445\pi\)
\(19\)	−0.509844	−	0.0982645i	−0.116966	−	0.0225434i	0.130432	−	0.991457i	\(-0.458364\pi\)
							−0.247398	+	0.968914i	\(0.579576\pi\)
\(23\)	2.81332	−	3.88397i	0.586617	−	0.809865i
							\(29\)	1.27636	+	1.47300i	0.237014	+	0.273529i	0.861779	−	0.507284i	\(-0.169351\pi\)
−0.624765	+	0.780813i	\(0.714805\pi\)
\(31\)	5.23124	−	4.98798i	0.939559	−	0.895867i	−0.0551631	−	0.998477i	\(-0.517568\pi\)
							0.994722	+	0.102610i	\(0.0327194\pi\)
\(37\)	6.73990	+	3.47466i	1.10803	+	0.571231i	0.912340	−	0.409434i	\(-0.134274\pi\)
							0.195693	+	0.980665i	\(0.437304\pi\)
\(41\)	4.89683	+	3.14700i	0.764756	+	0.491479i	0.863943	−	0.503590i	\(-0.167988\pi\)
							−0.0991865	+	0.995069i	\(0.531624\pi\)
\(43\)	0.302936	−	0.0889501i	0.0461973	−	0.0135648i	−0.258552	−	0.965997i	\(-0.583245\pi\)
							0.304750	+	0.952433i	\(0.401427\pi\)
\(47\)	1.27655	+	2.21106i	0.186204	+	0.322516i	0.943982	−	0.329998i	\(-0.107048\pi\)
							−0.757777	+	0.652513i	\(0.773715\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.4.3	✓	320
7.2	even	3	inner	483.2.y.b.142.14	yes	320
23.6	even	11	inner	483.2.y.b.466.14	yes	320
161.121	even	33	inner	483.2.y.b.121.3	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.4.3	✓	320	1.1	even	1	trivial
483.2.y.b.121.3	yes	320	161.121	even	33	inner
483.2.y.b.142.14	yes	320	7.2	even	3	inner
483.2.y.b.466.14	yes	320	23.6	even	11	inner