Embedded newform 483.2.u.a.113.1

• Label: 483.2.u.a.113.1
• Level: $483$
• Weight: $2$
• Character: 483.113
• Analytic conductor: $3.857$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.u (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(48\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			113.1
Character	\(\chi\)	\(=\)	483.113
Dual form			483.2.u.a.218.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.50106 + 2.33570i) q^{2} +(0.521192 - 1.65177i) q^{3} +(-2.37146 - 5.19277i) q^{4} +(-1.06257 - 0.311999i) q^{5} +(3.07570 + 3.69676i) q^{6} +(0.755750 + 0.654861i) q^{7} +(10.1921 + 1.46540i) q^{8} +(-2.45672 - 1.72178i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q + 4 q^{3} + 40 q^{4} - 6 q^{6} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{13}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.50106	+	2.33570i	−1.06141	+	1.65159i	−0.370163	+	0.928967i	\(0.620698\pi\)
							−0.691246	+	0.722619i	\(0.742938\pi\)
\(3\)	0.521192	−	1.65177i	0.300910	−	0.953652i
							\(5\)	−1.06257	−	0.311999i	−0.475196	−	0.139530i	0.0353602	−	0.999375i	\(-0.488742\pi\)
−0.510556	+	0.859845i	\(0.670560\pi\)
\(7\)	0.755750	+	0.654861i	0.285646	+	0.247514i
							\(11\)	0.907615	−	0.583289i	0.273656	−	0.175868i	−0.396614	−	0.917986i	\(-0.629815\pi\)
0.670270	+	0.742118i	\(0.266178\pi\)
\(13\)	−2.20919	−	2.54955i	−0.612720	−	0.707117i	0.361587	−	0.932338i	\(-0.382235\pi\)
							−0.974308	+	0.225221i	\(0.927690\pi\)
\(17\)	−2.23828	+	4.90114i	−0.542862	+	1.18870i	0.417175	+	0.908826i	\(0.363020\pi\)
							−0.960036	+	0.279875i	\(0.909707\pi\)
\(19\)	1.15636	−	0.528092i	0.265287	−	0.121153i	−0.278331	−	0.960485i	\(-0.589781\pi\)
							0.543618	+	0.839333i	\(0.317054\pi\)
\(23\)	−4.70697	−	0.918925i	−0.981471	−	0.191609i
							\(29\)	−6.23438	−	2.84715i	−1.15770	−	0.528702i	−0.258397	−	0.966039i	\(-0.583194\pi\)
−0.899298	+	0.437337i	\(0.855922\pi\)
\(31\)	0.712312	−	4.95424i	0.127935	−	0.889807i	−0.820231	−	0.572032i	\(-0.806155\pi\)
							0.948166	−	0.317775i	\(-0.102936\pi\)
\(37\)	−0.615380	−	2.09579i	−0.101168	−	0.344546i	0.893317	−	0.449427i	\(-0.148372\pi\)
							−0.994485	+	0.104881i	\(0.966554\pi\)
\(41\)	0.354450	−	1.20715i	0.0553558	−	0.188524i	−0.927174	−	0.374632i	\(-0.877769\pi\)
							0.982529	+	0.186107i	\(0.0595873\pi\)
\(43\)	3.70337	−	0.532464i	0.564758	−	0.0812000i	0.145978	−	0.989288i	\(-0.453367\pi\)
							0.418780	+	0.908088i	\(0.362458\pi\)
\(47\)			0.223569i			0.0326108i	0.999867	+	0.0163054i	\(0.00519041\pi\)
							−0.999867	+	0.0163054i	\(0.994810\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.u.a.113.1	✓	480
3.2	odd	2	inner	483.2.u.a.113.48	yes	480
23.11	odd	22	inner	483.2.u.a.218.48	yes	480
69.11	even	22	inner	483.2.u.a.218.1	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.u.a.113.1	✓	480	1.1	even	1	trivial
483.2.u.a.113.48	yes	480	3.2	odd	2	inner
483.2.u.a.218.1	yes	480	69.11	even	22	inner
483.2.u.a.218.48	yes	480	23.11	odd	22	inner