Embedded newform 483.2.y.b.121.3

• Label: 483.2.y.b.121.3
• Level: $483$
• Weight: $2$
• Character: 483.121
• 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			121.3
Character	\(\chi\)	\(=\)	483.121
Dual form			483.2.y.b.4.3

$q$-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 \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{9}{11}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.4.3	✓	320	161.4	even	33	inner
483.2.y.b.121.3	yes	320	1.1	even	1	trivial
483.2.y.b.142.14	yes	320	23.4	even	11	inner
483.2.y.b.466.14	yes	320	7.4	even	3	inner