Embedded newform 322.2.m.b.193.6

• Label: 322.2.m.b.193.6
• Level: $322$
• Weight: $2$
• Character: 322.193
• Analytic conductor: $2.571$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 322 = 2 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	322.m (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.57118294509\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(8\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			193.6
Character	\(\chi\)	\(=\)	322.193
Dual form			322.2.m.b.317.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.723734 + 0.690079i) q^{2} +(1.57976 - 0.304474i) q^{3} +(0.0475819 + 0.998867i) q^{4} +(1.18117 - 0.928883i) q^{5} +(1.35344 + 0.869801i) q^{6} +(2.64401 - 0.0960643i) q^{7} +(-0.654861 + 0.755750i) q^{8} +(-0.382166 + 0.152996i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{2} - 2 q^{3} + 8 q^{4} + 2 q^{5} + 4 q^{6} - 11 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.723734	+	0.690079i	0.511757	+	0.487960i
							\(3\)	1.57976	−	0.304474i	0.912075	−	0.175788i	0.288435	−	0.957500i	\(-0.406865\pi\)
0.623640	+	0.781712i	\(0.285653\pi\)
\(5\)	1.18117	−	0.928883i	0.528235	−	0.415409i	−0.318102	−	0.948057i	\(-0.603045\pi\)
							0.846337	+	0.532648i	\(0.178803\pi\)
\(7\)	2.64401	−	0.0960643i	0.999341	−	0.0363089i
							\(11\)	−3.07218	+	2.92932i	−0.926298	+	0.883224i	−0.993421	−	0.114521i	\(-0.963467\pi\)
0.0671226	+	0.997745i	\(0.478618\pi\)
\(13\)	−2.57194	−	5.63176i	−0.713327	−	1.56197i	−0.823026	−	0.568004i	\(-0.807716\pi\)
							0.109698	−	0.993965i	\(-0.465012\pi\)
\(17\)	1.27605	−	0.657849i	0.309487	−	0.159552i	−0.296485	−	0.955037i	\(-0.595814\pi\)
							0.605972	+	0.795486i	\(0.292784\pi\)
\(19\)	−1.55789	−	0.803147i	−0.357404	−	0.184254i	0.270168	−	0.962813i	\(-0.412921\pi\)
							−0.627572	+	0.778559i	\(0.715951\pi\)
\(23\)	4.33646	+	2.04819i	0.904215	+	0.427077i
							\(29\)	1.35187	+	0.868793i	0.251036	+	0.161331i	0.660107	−	0.751172i	\(-0.270511\pi\)
−0.409071	+	0.912502i	\(0.634147\pi\)
\(31\)	0.231235	+	0.668110i	0.0415311	+	0.119996i	0.963860	−	0.266410i	\(-0.0858375\pi\)
							−0.922329	+	0.386406i	\(0.873716\pi\)
\(37\)	−5.54271	+	2.21897i	−0.911217	+	0.364796i	−0.779435	−	0.626483i	\(-0.784494\pi\)
							−0.131781	+	0.991279i	\(0.542070\pi\)
\(41\)	0.620949	+	4.31880i	0.0969760	+	0.674483i	0.979087	+	0.203441i	\(0.0652126\pi\)
							−0.882111	+	0.471041i	\(0.843878\pi\)
\(43\)	−6.27143	−	7.23762i	−0.956384	−	1.10373i	−0.994530	−	0.104454i	\(-0.966691\pi\)
							0.0381457	−	0.999272i	\(-0.487855\pi\)
\(47\)	−2.01036	−	3.48205i	−0.293241	−	0.507909i	0.681333	−	0.731974i	\(-0.261401\pi\)
							−0.974574	+	0.224065i	\(0.928067\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.m.b.193.6	yes	160
7.2	even	3	inner	322.2.m.b.9.3	✓	160
23.18	even	11	inner	322.2.m.b.179.3	yes	160
161.156	even	33	inner	322.2.m.b.317.6	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.m.b.9.3	✓	160	7.2	even	3	inner
322.2.m.b.179.3	yes	160	23.18	even	11	inner
322.2.m.b.193.6	yes	160	1.1	even	1	trivial
322.2.m.b.317.6	yes	160	161.156	even	33	inner