Embedded newform 161.2.o.a.10.10

• Label: 161.2.o.a.10.10
• Level: $161$
• Weight: $2$
• Character: 161.10
• Analytic conductor: $1.286$
• Analytic rank: $0$
• Dimension: $280$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 161 = 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	161.o (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.28559147254\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(14\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			10.10
Character	\(\chi\)	\(=\)	161.10
Dual form			161.2.o.a.145.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.352116 - 1.01737i) q^{2} +(1.36243 + 2.64275i) q^{3} +(0.661046 + 0.519852i) q^{4} +(-1.73069 + 0.165261i) q^{5} +(3.16839 - 0.455546i) q^{6} +(-1.93398 - 1.80547i) q^{7} +(2.57301 - 1.65357i) q^{8} +(-3.38774 + 4.75741i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 7 q^{2} - 27 q^{3} + q^{4} - 33 q^{5} - 22 q^{7} - 56 q^{8} - 15 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/161\mathbb{Z}\right)^\times\).

\(n\)	\(24\)	\(120\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.352116	−	1.01737i	0.248983	−	0.719391i	−0.749377	−	0.662144i	\(-0.769647\pi\)
							0.998360	−	0.0572465i	\(-0.0182321\pi\)
\(3\)	1.36243	+	2.64275i	0.786601	+	1.52579i	0.849390	+	0.527766i	\(0.176970\pi\)
							−0.0627893	+	0.998027i	\(0.520000\pi\)
\(5\)	−1.73069	+	0.165261i	−0.773989	+	0.0739070i	−0.474574	−	0.880216i	\(-0.657398\pi\)
							−0.299416	+	0.954123i	\(0.596792\pi\)
\(7\)	−1.93398	−	1.80547i	−0.730975	−	0.682404i
							\(11\)	2.98085	−	1.03168i	0.898760	−	0.311064i	0.161659	−	0.986847i	\(-0.448315\pi\)
0.737101	+	0.675783i	\(0.236194\pi\)
\(13\)	0.572313	+	1.94912i	0.158731	+	0.540588i	1.00000		2.84148e-5i	\(-9.04470e-6\pi\)
							−0.841269	+	0.540617i	\(0.818191\pi\)
\(17\)	−4.10066	−	1.64166i	−0.994555	−	0.398160i	−0.183430	−	0.983033i	\(-0.558720\pi\)
							−0.811125	+	0.584873i	\(0.801144\pi\)
\(19\)	3.08458	−	1.23488i	0.707651	−	0.283301i	0.0102153	−	0.999948i	\(-0.496748\pi\)
							0.697436	+	0.716647i	\(0.254324\pi\)
\(23\)	1.68397	−	4.49046i	0.351132	−	0.936326i
							\(29\)	−1.34849	−	9.37896i	−0.250409	−	1.74163i	−0.595766	−	0.803158i	\(-0.703151\pi\)
0.345357	−	0.938471i	\(-0.387758\pi\)
\(31\)	−5.71624	+	0.272298i	−1.02667	+	0.0489062i	−0.554151	−	0.832416i	\(-0.686957\pi\)
							−0.472516	+	0.881322i	\(0.656654\pi\)
\(37\)	1.10208	+	0.784787i	0.181181	+	0.129018i	0.667049	−	0.745014i	\(-0.267557\pi\)
							−0.485868	+	0.874032i	\(0.661497\pi\)
\(41\)	6.62922	+	3.02747i	1.03531	+	0.472811i	0.859243	−	0.511567i	\(-0.170935\pi\)
							0.176068	+	0.984378i	\(0.443662\pi\)
\(43\)	−4.55880	+	7.09363i	−0.695210	+	1.08177i	0.296718	+	0.954965i	\(0.404108\pi\)
							−0.991928	+	0.126803i	\(0.959528\pi\)
\(47\)	−5.84355	+	3.37378i	−0.852370	+	0.492116i	−0.861450	−	0.507843i	\(-0.830443\pi\)
							0.00907992	+	0.999959i	\(0.497110\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	161.2.o.a.10.10	✓	280
7.5	odd	6	inner	161.2.o.a.33.5	yes	280
23.7	odd	22	inner	161.2.o.a.122.5	yes	280
161.145	even	66	inner	161.2.o.a.145.10	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.o.a.10.10	✓	280	1.1	even	1	trivial
161.2.o.a.33.5	yes	280	7.5	odd	6	inner
161.2.o.a.122.5	yes	280	23.7	odd	22	inner
161.2.o.a.145.10	yes	280	161.145	even	66	inner