Embedded newform 161.2.o.a.10.7

• Label: 161.2.o.a.10.7
• 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			10.7
Character	\(\chi\)	\(=\)	161.10
Dual form			161.2.o.a.145.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0473784 + 0.136891i) q^{2} +(-0.509808 - 0.988890i) q^{3} +(1.55561 + 1.22335i) q^{4} +(0.991613 - 0.0946875i) q^{5} +(0.159524 - 0.0229361i) q^{6} +(-0.848038 + 2.50616i) q^{7} +(-0.484892 + 0.311621i) q^{8} +(1.02217 - 1.43544i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 7 q^{2} - 27 q^{3} + q^{4} - 33 q^{5} - 22 q^{7} - 56 q^{8} - 15 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0473784	+	0.136891i	−0.0335016	+	0.0967964i	−0.960511	−	0.278241i	\(-0.910249\pi\)
							0.927010	+	0.375037i	\(0.122370\pi\)
\(3\)	−0.509808	−	0.988890i	−0.294338	−	0.570936i	0.694577	−	0.719418i	\(-0.255591\pi\)
							−0.988915	+	0.148482i	\(0.952561\pi\)
\(5\)	0.991613	−	0.0946875i	0.443463	−	0.0423456i	0.129063	−	0.991636i	\(-0.458803\pi\)
							0.314399	+	0.949291i	\(0.398197\pi\)
\(7\)	−0.848038	+	2.50616i	−0.320528	+	0.947239i
							\(11\)	3.11439	−	1.07790i	0.939022	−	0.324999i	0.185667	−	0.982613i	\(-0.440555\pi\)
0.753355	+	0.657614i	\(0.228434\pi\)
\(13\)	−0.221566	−	0.754586i	−0.0614515	−	0.209285i	0.923044	−	0.384694i	\(-0.125693\pi\)
							−0.984496	+	0.175410i	\(0.943875\pi\)
\(17\)	−4.01245	−	1.60634i	−0.973163	−	0.389596i	−0.170058	−	0.985434i	\(-0.554396\pi\)
							−0.803104	+	0.595838i	\(0.796820\pi\)
\(19\)	2.45254	−	0.981851i	0.562652	−	0.225252i	−0.0728608	−	0.997342i	\(-0.523213\pi\)
							0.635513	+	0.772090i	\(0.280789\pi\)
\(23\)	2.00364	+	4.35723i	0.417789	+	0.908544i
							\(29\)	−1.03403	−	7.19183i	−0.192014	−	1.33549i	−0.826667	−	0.562691i	\(-0.809766\pi\)
0.634653	−	0.772797i	\(-0.281143\pi\)
\(31\)	−8.24968	+	0.392981i	−1.48169	+	0.0705814i	−0.772759	−	0.634700i	\(-0.781124\pi\)
							−0.708928	+	0.705281i	\(0.750821\pi\)
\(37\)	−0.700524	−	0.498841i	−0.115165	−	0.0820089i	0.521023	−	0.853543i	\(-0.325551\pi\)
							−0.636188	+	0.771534i	\(0.719490\pi\)
\(41\)	−5.91907	−	2.70315i	−0.924403	−	0.422161i	−0.104410	−	0.994534i	\(-0.533295\pi\)
							−0.819993	+	0.572374i	\(0.806023\pi\)
\(43\)	−5.83894	+	9.08558i	−0.890431	+	1.38554i	0.0320468	+	0.999486i	\(0.489797\pi\)
							−0.922477	+	0.386051i	\(0.873839\pi\)
\(47\)	7.88048	−	4.54980i	1.14949	−	0.663656i	0.200724	−	0.979648i	\(-0.435670\pi\)
							0.948762	+	0.315991i	\(0.102337\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	161.2.o.a.10.7	✓	280
7.5	odd	6	inner	161.2.o.a.33.8	yes	280
23.7	odd	22	inner	161.2.o.a.122.8	yes	280
161.145	even	66	inner	161.2.o.a.145.7	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.o.a.10.7	✓	280	1.1	even	1	trivial
161.2.o.a.33.8	yes	280	7.5	odd	6	inner
161.2.o.a.122.8	yes	280	23.7	odd	22	inner
161.2.o.a.145.7	yes	280	161.145	even	66	inner