Embedded newform 322.2.o.a.227.6

• Label: 322.2.o.a.227.6
• Level: $322$
• Weight: $2$
• Character: 322.227
• 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.o (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.57118294509\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(8\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			227.6
Character	\(\chi\)	\(=\)	322.227
Dual form			322.2.o.a.61.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.786053 - 0.618159i) q^{2} +(1.31158 - 0.933973i) q^{3} +(0.235759 - 0.971812i) q^{4} +(1.08406 + 0.208935i) q^{5} +(0.453629 - 1.54492i) q^{6} +(0.523511 + 2.59344i) q^{7} +(-0.415415 - 0.909632i) q^{8} +(-0.133263 + 0.385038i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 6 q^{3} + 8 q^{4} - 11 q^{7} + 16 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.786053	−	0.618159i	0.555823	−	0.437104i
							\(3\)	1.31158	−	0.933973i	0.757242	−	0.539230i	−0.134968	−	0.990850i	\(-0.543093\pi\)
0.892210	+	0.451620i	\(0.149154\pi\)
\(5\)	1.08406	+	0.208935i	0.484805	+	0.0934386i	0.425797	−	0.904819i	\(-0.359994\pi\)
							0.0590087	+	0.998257i	\(0.481206\pi\)
\(7\)	0.523511	+	2.59344i	0.197868	+	0.980229i
							\(11\)	1.95431	−	2.48510i	0.589246	−	0.749287i	−0.396407	−	0.918075i	\(-0.629743\pi\)
0.985652	+	0.168788i	\(0.0539854\pi\)
\(13\)	0.970556	−	1.51022i	0.269184	−	0.418858i	−0.680176	−	0.733049i	\(-0.738097\pi\)
							0.949360	+	0.314190i	\(0.101733\pi\)
\(17\)	−2.88365	+	2.74956i	−0.699389	+	0.666866i	−0.953873	−	0.300211i	\(-0.902943\pi\)
							0.254484	+	0.967077i	\(0.418094\pi\)
\(19\)	−4.50054	−	4.29125i	−1.03249	−	0.984481i	−0.0326066	−	0.999468i	\(-0.510381\pi\)
							−0.999888	+	0.0149869i	\(0.995229\pi\)
\(23\)	−4.03153	+	2.59746i	−0.840632	+	0.541607i
							\(29\)	−6.53137	−	1.91778i	−1.21285	−	0.356124i	−0.388094	−	0.921620i	\(-0.626867\pi\)
−0.824751	+	0.565496i	\(0.808685\pi\)
\(31\)	0.0568353	−	0.595206i	0.0102079	−	0.106902i	−0.988932	−	0.148370i	\(-0.952597\pi\)
							0.999140	+	0.0414678i	\(0.0132034\pi\)
\(37\)	8.52060	+	2.94901i	1.40078	+	0.484814i	0.919962	−	0.392007i	\(-0.128219\pi\)
							0.480816	+	0.876821i	\(0.340340\pi\)
\(41\)	0.356885	+	0.309243i	0.0557361	+	0.0482956i	0.682283	−	0.731088i	\(-0.260987\pi\)
							−0.626547	+	0.779383i	\(0.715532\pi\)
\(43\)	9.92714	+	4.53357i	1.51387	+	0.691363i	0.987315	−	0.158775i	\(-0.0507545\pi\)
							0.526559	+	0.850138i	\(0.323482\pi\)
\(47\)	−0.473951	+	0.273636i	−0.0691329	+	0.0399139i	−0.534168	−	0.845378i	\(-0.679375\pi\)
							0.465035	+	0.885292i	\(0.346042\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.a.227.6	yes	160
7.5	odd	6	inner	322.2.o.a.89.3	yes	160
23.15	odd	22	inner	322.2.o.a.199.3	yes	160
161.61	even	66	inner	322.2.o.a.61.6	✓	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.a.61.6	✓	160	161.61	even	66	inner
322.2.o.a.89.3	yes	160	7.5	odd	6	inner
322.2.o.a.199.3	yes	160	23.15	odd	22	inner
322.2.o.a.227.6	yes	160	1.1	even	1	trivial