Embedded newform 161.2.m.a.100.2

• Label: 161.2.m.a.100.2
• Level: $161$
• Weight: $2$
• Character: 161.100
• 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.m (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.28559147254\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(14\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			100.2
Character	\(\chi\)	\(=\)	161.100
Dual form			161.2.m.a.95.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.59763 - 1.25639i) q^{2} +(1.76169 - 2.47395i) q^{3} +(0.502387 + 2.07087i) q^{4} +(-2.84890 + 0.549081i) q^{5} +(-5.92278 + 1.73908i) q^{6} +(-2.63294 + 0.260035i) q^{7} +(0.110549 - 0.242070i) q^{8} +(-2.03567 - 5.88168i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 11 q^{2} - 9 q^{3} + q^{4} - 11 q^{5} - 52 q^{6} - 20 q^{7} - 32 q^{8} + 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(24\)	\(120\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{3}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.59763	−	1.25639i	−1.12969	−	0.888402i	−0.134956	−	0.990852i	\(-0.543089\pi\)
							−0.994738	+	0.102450i	\(0.967332\pi\)
\(3\)	1.76169	−	2.47395i	1.01711	−	1.42834i	0.116769	−	0.993159i	\(-0.462746\pi\)
							0.900345	−	0.435178i	\(-0.143314\pi\)
\(5\)	−2.84890	+	0.549081i	−1.27407	+	0.245557i	−0.781054	−	0.624464i	\(-0.785317\pi\)
							−0.493015	+	0.870021i	\(0.664105\pi\)
\(7\)	−2.63294	+	0.260035i	−0.995158	+	0.0982841i
							\(11\)	0.552710	−	0.434656i	0.166648	−	0.131054i	−0.531352	−	0.847151i	\(-0.678316\pi\)
0.698000	+	0.716098i	\(0.254073\pi\)
\(13\)	3.04838	−	1.95907i	0.845468	−	0.543349i	−0.0446901	−	0.999001i	\(-0.514230\pi\)
							0.890158	+	0.455652i	\(0.150594\pi\)
\(17\)	−3.27094	−	3.11883i	−0.793318	−	0.756428i	0.180544	−	0.983567i	\(-0.442214\pi\)
							−0.973862	+	0.227139i	\(0.927063\pi\)
\(19\)	2.90436	−	2.76930i	0.666305	−	0.635321i	−0.279466	−	0.960156i	\(-0.590157\pi\)
							0.945771	+	0.324835i	\(0.105309\pi\)
\(23\)	−4.72195	−	0.838585i	−0.984594	−	0.174857i
							\(29\)	8.16957	−	2.39880i	1.51705	−	0.445446i	0.585992	−	0.810317i	\(-0.300705\pi\)
0.931058	+	0.364870i	\(0.118887\pi\)
\(31\)	−0.658512	+	0.0628802i	−0.118272	+	0.0112936i	−0.154024	−	0.988067i	\(-0.549223\pi\)
							0.0357521	+	0.999361i	\(0.488617\pi\)
\(37\)	2.87836	+	8.31646i	0.473199	+	1.36722i	0.889060	+	0.457790i	\(0.151359\pi\)
							−0.415861	+	0.909428i	\(0.636520\pi\)
\(41\)	−0.195405	−	0.225510i	−0.0305172	−	0.0352187i	0.740286	−	0.672292i	\(-0.234690\pi\)
							−0.770803	+	0.637074i	\(0.780145\pi\)
\(43\)	−2.93242	−	6.42111i	−0.447190	−	0.979210i	−0.990222	−	0.139498i	\(-0.955451\pi\)
							0.543032	−	0.839712i	\(-0.317276\pi\)
\(47\)	3.07014	−	5.31764i	0.447826	−	0.775658i	−0.550418	−	0.834889i	\(-0.685532\pi\)
							0.998244	+	0.0592315i	\(0.0188650\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	161.2.m.a.100.2	yes	280
7.4	even	3	inner	161.2.m.a.123.13	yes	280
23.3	even	11	inner	161.2.m.a.72.13	✓	280
161.95	even	33	inner	161.2.m.a.95.2	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.m.a.72.13	✓	280	23.3	even	11	inner
161.2.m.a.95.2	yes	280	161.95	even	33	inner
161.2.m.a.100.2	yes	280	1.1	even	1	trivial
161.2.m.a.123.13	yes	280	7.4	even	3	inner