Embedded newform 483.2.y.b.361.13

• Label: 483.2.y.b.361.13
• Level: $483$
• Weight: $2$
• Character: 483.361
• 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			361.13
Character	\(\chi\)	\(=\)	483.361
Dual form			483.2.y.b.289.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.86410 + 0.359276i) q^{2} +(0.0475819 + 0.998867i) q^{3} +(1.48907 + 0.596132i) q^{4} +(1.84600 - 2.59234i) q^{5} +(-0.270172 + 1.87909i) q^{6} +(2.39172 + 1.13122i) q^{7} +(-0.632491 - 0.406478i) q^{8} +(-0.995472 + 0.0950560i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{4}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.86410	+	0.359276i	1.31812	+	0.254047i	0.799349	−	0.600867i	\(-0.205178\pi\)
							0.518771	+	0.854913i	\(0.326390\pi\)
\(3\)	0.0475819	+	0.998867i	0.0274714	+	0.576696i
							\(5\)	1.84600	−	2.59234i	0.825555	−	1.15933i	−0.159518	−	0.987195i	\(-0.550994\pi\)
0.985073	−	0.172135i	\(-0.0550666\pi\)
\(7\)	2.39172	+	1.13122i	0.903986	+	0.427561i
							\(11\)	2.53066	−	0.487746i	0.763024	−	0.147061i	0.207129	−	0.978314i	\(-0.433588\pi\)
0.555895	+	0.831253i	\(0.312376\pi\)
\(13\)	−3.50062	−	1.02787i	−0.970897	−	0.285081i	−0.242436	−	0.970168i	\(-0.577946\pi\)
							−0.728462	+	0.685086i	\(0.759764\pi\)
\(17\)	1.21426	+	0.954905i	0.294502	+	0.231599i	0.754467	−	0.656338i	\(-0.227896\pi\)
							−0.459965	+	0.887937i	\(0.652138\pi\)
\(19\)	−2.30239	+	1.81062i	−0.528205	+	0.415385i	−0.846326	−	0.532665i	\(-0.821191\pi\)
							0.318121	+	0.948050i	\(0.396948\pi\)
\(23\)	1.78897	+	4.44967i	0.373026	+	0.927821i
							\(29\)	−1.35096	+	9.39616i	−0.250868	+	1.74482i	0.342161	+	0.939641i	\(0.388841\pi\)
−0.593028	+	0.805182i	\(0.702068\pi\)
\(31\)	0.442501	−	0.228125i	0.0794756	−	0.0409725i	−0.418031	−	0.908433i	\(-0.637280\pi\)
							0.497506	+	0.867460i	\(0.334249\pi\)
\(37\)	−7.43133	+	0.709606i	−1.22170	+	0.116659i	−0.685908	−	0.727689i	\(-0.740595\pi\)
							−0.535796	+	0.844347i	\(0.679988\pi\)
\(41\)	−3.04181	−	6.66063i	−0.475050	−	1.04022i	−0.983795	−	0.179298i	\(-0.942618\pi\)
							0.508745	−	0.860917i	\(-0.330110\pi\)
\(43\)	2.95900	−	1.90163i	0.451243	−	0.289996i	−0.295213	−	0.955432i	\(-0.595390\pi\)
							0.746455	+	0.665435i	\(0.231754\pi\)
\(47\)	−4.58771	−	7.94615i	−0.669186	−	1.15906i	−0.978132	−	0.207985i	\(-0.933309\pi\)
							0.308946	−	0.951080i	\(-0.400024\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.361.13	yes	320
7.2	even	3	inner	483.2.y.b.16.4	✓	320
23.13	even	11	inner	483.2.y.b.151.4	yes	320
161.128	even	33	inner	483.2.y.b.289.13	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.16.4	✓	320	7.2	even	3	inner
483.2.y.b.151.4	yes	320	23.13	even	11	inner
483.2.y.b.289.13	yes	320	161.128	even	33	inner
483.2.y.b.361.13	yes	320	1.1	even	1	trivial