Embedded newform 322.2.m.a.81.5

• Label: 322.2.m.a.81.5
• Level: $322$
• Weight: $2$
• Character: 322.81
• 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			81.5
Character	\(\chi\)	\(=\)	322.81
Dual form			322.2.m.a.163.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.888835 + 0.458227i) q^{2} +(0.400612 + 0.315045i) q^{3} +(0.580057 + 0.814576i) q^{4} +(3.05166 + 2.90976i) q^{5} +(0.211716 + 0.463594i) q^{6} +(-0.375580 - 2.61896i) q^{7} +(0.142315 + 0.989821i) q^{8} +(-0.646040 - 2.66301i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 4 q^{6} + 15 q^{7} + 16 q^{8} + 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{10}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.888835	+	0.458227i	0.628502	+	0.324015i
							\(3\)	0.400612	+	0.315045i	0.231294	+	0.181891i	0.727079	−	0.686554i	\(-0.240877\pi\)
−0.495785	+	0.868445i	\(0.665120\pi\)
\(5\)	3.05166	+	2.90976i	1.36475	+	1.30128i	0.912319	+	0.409481i	\(0.134290\pi\)
							0.452427	+	0.891802i	\(0.350558\pi\)
\(7\)	−0.375580	−	2.61896i	−0.141956	−	0.989873i
							\(11\)	−0.296482	+	0.152847i	−0.0893925	+	0.0460851i	−0.502343	−	0.864668i	\(-0.667529\pi\)
0.412951	+	0.910753i	\(0.364498\pi\)
\(13\)	−1.63847	+	1.89090i	−0.454430	+	0.524441i	−0.936016	−	0.351959i	\(-0.885516\pi\)
							0.481585	+	0.876399i	\(0.340061\pi\)
\(17\)	−5.32210	−	0.508199i	−1.29080	−	0.123256i	−0.573028	−	0.819536i	\(-0.694231\pi\)
							−0.717771	+	0.696279i	\(0.754837\pi\)
\(19\)	3.18253	−	0.303895i	0.730123	−	0.0697183i	0.276628	−	0.960977i	\(-0.410783\pi\)
							0.453495	+	0.891259i	\(0.350177\pi\)
\(23\)	−2.87613	−	3.83769i	−0.599715	−	0.800214i
							\(29\)	−2.04548	−	4.47898i	−0.379836	−	0.831725i	−0.998923	−	0.0464025i	\(-0.985224\pi\)
0.619087	−	0.785323i	\(-0.287503\pi\)
\(31\)	2.97187	+	1.18976i	0.533764	+	0.213687i	0.622856	−	0.782336i	\(-0.285972\pi\)
							−0.0890918	+	0.996023i	\(0.528396\pi\)
\(37\)	1.79862	+	7.41402i	0.295692	+	1.21886i	0.904823	+	0.425788i	\(0.140003\pi\)
							−0.609131	+	0.793069i	\(0.708482\pi\)
\(41\)	5.96414	−	1.75123i	0.931442	−	0.273496i	0.219402	−	0.975635i	\(-0.429590\pi\)
							0.712040	+	0.702139i	\(0.247771\pi\)
\(43\)	0.573164	−	3.98644i	0.0874067	−	0.607927i	−0.898291	−	0.439402i	\(-0.855190\pi\)
							0.985697	−	0.168525i	\(-0.0539005\pi\)
\(47\)	0.0536261	+	0.0928832i	0.00782218	+	0.0135484i	0.869910	−	0.493210i	\(-0.164177\pi\)
							−0.862088	+	0.506759i	\(0.830843\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.m.a.81.5	yes	160
7.2	even	3	inner	322.2.m.a.219.4	yes	160
23.2	even	11	inner	322.2.m.a.25.4	✓	160
161.2	even	33	inner	322.2.m.a.163.5	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.m.a.25.4	✓	160	23.2	even	11	inner
322.2.m.a.81.5	yes	160	1.1	even	1	trivial
322.2.m.a.163.5	yes	160	161.2	even	33	inner
322.2.m.a.219.4	yes	160	7.2	even	3	inner