Embedded newform 162.10.c.i.109.1

• Label: 162.10.c.i.109.1
• Level: $162$
• Weight: $10$
• Character: 162.109
• Analytic conductor: $83.436$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	162.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(83.4358054585\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 2)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			\(0.500000 - 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	162.109
Dual form			162.10.c.i.55.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(8.00000 + 13.8564i) q^{2} +(-128.000 + 221.703i) q^{4} +(435.000 - 753.442i) q^{5} +(476.000 + 824.456i) q^{7} -4096.00 q^{8} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 16 q^{2} - 256 q^{4} + 870 q^{5} + 952 q^{7} - 8192 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\)	\(83\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	8.00000	+	13.8564i	0.353553	+	0.612372i
							\(3\)		0			0
\(5\)	435.000	−	753.442i	0.311261	−	0.539119i								−0.667375	−	0.744722i	\(-0.732582\pi\)
							0.978636	+	0.205603i	\(0.0659154\pi\)
\(7\)	476.000	+	824.456i	0.0749317	+	0.129786i	0.901057	−	0.433702i	\(-0.142793\pi\)
							−0.826125	+	0.563487i	\(0.809459\pi\)
\(11\)	−28074.0	−	48625.6i	−0.578146	−	1.00138i	−0.995692	−	0.0927219i	\(-0.970443\pi\)
							0.417546	−	0.908656i	\(-0.362890\pi\)
\(13\)	−89047.0	+	154234.i	−0.864717	+	1.49773i	0.00261007	+	0.999997i	\(0.499169\pi\)
							−0.867327	+	0.497738i	\(0.834164\pi\)
\(17\)	247662.			0.719183			0.359591	−	0.933110i	\(-0.382916\pi\)
							0.359591	+	0.933110i	\(0.382916\pi\)
\(19\)	315380.			0.555192			0.277596	−	0.960698i	\(-0.410462\pi\)
							0.277596	+	0.960698i	\(0.410462\pi\)
\(23\)	102252.	−	177106.i	0.0761898	−	0.131965i	−0.825413	−	0.564529i	\(-0.809058\pi\)
							0.901603	+	0.432564i	\(0.142391\pi\)
\(29\)	−1.92023e6	−	3.32593e6i	−0.504152	−	0.873216i	−0.999988	−	0.00480048i	\(-0.998472\pi\)
							0.495837	−	0.868416i	\(-0.334861\pi\)
\(31\)	654704.	−	1.13398e6i	0.127326	−	0.220535i	−0.795314	−	0.606198i	\(-0.792694\pi\)
							0.922640	+	0.385663i	\(0.126027\pi\)
\(37\)	4.30708e6			0.377811			0.188906	−	0.981995i	\(-0.439506\pi\)
							0.188906	+	0.981995i	\(0.439506\pi\)
\(41\)	756021.	−	1.30947e6i	0.0417837	−	0.0723714i	−0.844377	−	0.535749i	\(-0.820029\pi\)
							0.886161	+	0.463378i	\(0.153363\pi\)
\(43\)	−1.68353e7	−	2.91596e7i	−0.750953	−	1.30069i	−0.947361	−	0.320167i	\(-0.896261\pi\)
							0.196408	−	0.980522i	\(-0.437072\pi\)
\(47\)	−5.29054e6	−	9.16348e6i	−0.158146	−	0.273918i	0.776054	−	0.630667i	\(-0.217218\pi\)
							−0.934200	+	0.356749i	\(0.883885\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.10.c.i.109.1		2
3.2	odd	2		162.10.c.b.109.1		2
9.2	odd	6		162.10.c.b.55.1		2
9.4	even	3		18.10.a.a.1.1		1
9.5	odd	6		2.10.a.a.1.1	✓	1
9.7	even	3	inner	162.10.c.i.55.1		2
36.23	even	6		16.10.a.d.1.1		1
36.31	odd	6		144.10.a.d.1.1		1
45.14	odd	6		50.10.a.c.1.1		1
45.23	even	12		50.10.b.a.49.1		2
45.32	even	12		50.10.b.a.49.2		2
63.5	even	6		98.10.c.b.67.1		2
63.23	odd	6		98.10.c.c.67.1		2
63.32	odd	6		98.10.c.c.79.1		2
63.41	even	6		98.10.a.c.1.1		1
63.59	even	6		98.10.c.b.79.1		2
72.5	odd	6		64.10.a.h.1.1		1
72.59	even	6		64.10.a.b.1.1		1
99.32	even	6		242.10.a.a.1.1		1
117.77	odd	6		338.10.a.a.1.1		1
144.5	odd	12		256.10.b.g.129.2		2
144.59	even	12		256.10.b.e.129.1		2
144.77	odd	12		256.10.b.g.129.1		2
144.131	even	12		256.10.b.e.129.2		2
180.23	odd	12		400.10.c.d.49.2		2
180.59	even	6		400.10.a.b.1.1		1
180.167	odd	12		400.10.c.d.49.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.10.a.a.1.1	✓	1	9.5	odd	6
16.10.a.d.1.1		1	36.23	even	6
18.10.a.a.1.1		1	9.4	even	3
50.10.a.c.1.1		1	45.14	odd	6
50.10.b.a.49.1		2	45.23	even	12
50.10.b.a.49.2		2	45.32	even	12
64.10.a.b.1.1		1	72.59	even	6
64.10.a.h.1.1		1	72.5	odd	6
98.10.a.c.1.1		1	63.41	even	6
98.10.c.b.67.1		2	63.5	even	6
98.10.c.b.79.1		2	63.59	even	6
98.10.c.c.67.1		2	63.23	odd	6
98.10.c.c.79.1		2	63.32	odd	6
144.10.a.d.1.1		1	36.31	odd	6
162.10.c.b.55.1		2	9.2	odd	6
162.10.c.b.109.1		2	3.2	odd	2
162.10.c.i.55.1		2	9.7	even	3	inner
162.10.c.i.109.1		2	1.1	even	1	trivial
242.10.a.a.1.1		1	99.32	even	6
256.10.b.e.129.1		2	144.59	even	12
256.10.b.e.129.2		2	144.131	even	12
256.10.b.g.129.1		2	144.77	odd	12
256.10.b.g.129.2		2	144.5	odd	12
338.10.a.a.1.1		1	117.77	odd	6
400.10.a.b.1.1		1	180.59	even	6
400.10.c.d.49.1		2	180.167	odd	12
400.10.c.d.49.2		2	180.23	odd	12