Embedded newform 322.2.m.b.219.5

• Label: 322.2.m.b.219.5
• Level: $322$
• Weight: $2$
• Character: 322.219
• 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			219.5
Character	\(\chi\)	\(=\)	322.219
Dual form			322.2.m.b.25.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0475819 + 0.998867i) q^{2} +(0.0359061 - 0.0143747i) q^{3} +(-0.995472 + 0.0950560i) q^{4} +(-0.272928 + 1.12502i) q^{5} +(0.0160669 + 0.0351815i) q^{6} +(2.64127 - 0.153951i) q^{7} +(-0.142315 - 0.989821i) q^{8} +(-2.17012 + 2.06920i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{2} - 2 q^{3} + 8 q^{4} + 2 q^{5} + 4 q^{6} - 11 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{10}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0475819	+	0.998867i	0.0336455	+	0.706306i
							\(3\)	0.0359061	−	0.0143747i	0.0207304	−	0.00829921i	−0.361274	−	0.932460i	\(-0.617658\pi\)
0.382005	+	0.924160i	\(0.375234\pi\)
\(5\)	−0.272928	+	1.12502i	−0.122057	+	0.503125i	0.877659	+	0.479286i	\(0.159104\pi\)
							−0.999716	+	0.0238397i	\(0.992411\pi\)
\(7\)	2.64127	−	0.153951i	0.998306	−	0.0581881i
							\(11\)	−0.218504	+	4.58697i	−0.0658816	+	1.38302i	0.687855	+	0.725848i	\(0.258552\pi\)
−0.753737	+	0.657177i	\(0.771751\pi\)
\(13\)	2.27006	−	2.61979i	0.629602	−	0.726599i	−0.347899	−	0.937532i	\(-0.613105\pi\)
							0.977501	+	0.210933i	\(0.0676502\pi\)
\(17\)	−2.83882	+	3.98657i	−0.688516	+	0.966885i	0.311347	+	0.950296i	\(0.399220\pi\)
							−0.999862	+	0.0165885i	\(0.994719\pi\)
\(19\)	2.20755	+	3.10007i	0.506446	+	0.711204i	0.986081	−	0.166264i	\(-0.0531704\pi\)
							−0.479635	+	0.877468i	\(0.659231\pi\)
\(23\)	−4.29455	−	2.13467i	−0.895476	−	0.445109i
							\(29\)	−1.03836	−	2.27368i	−0.192818	−	0.422212i	0.788388	−	0.615179i	\(-0.210916\pi\)
−0.981205	+	0.192967i	\(0.938189\pi\)
\(31\)	−1.03274	+	0.812158i	−0.185486	+	0.145868i	−0.706584	−	0.707629i	\(-0.749765\pi\)
							0.521098	+	0.853497i	\(0.325523\pi\)
\(37\)	0.516519	−	0.492499i	0.0849151	−	0.0809664i	−0.646433	−	0.762971i	\(-0.723740\pi\)
							0.731348	+	0.682004i	\(0.238892\pi\)
\(41\)	5.92926	−	1.74099i	0.925995	−	0.271897i	0.216236	−	0.976341i	\(-0.430622\pi\)
							0.709759	+	0.704444i	\(0.248804\pi\)
\(43\)	0.625977	−	4.35376i	0.0954605	−	0.663943i	−0.884762	−	0.466043i	\(-0.845679\pi\)
							0.980222	−	0.197899i	\(-0.0634119\pi\)
\(47\)	2.98537	−	5.17080i	0.435460	−	0.754239i	−0.561873	−	0.827224i	\(-0.689919\pi\)
							0.997333	+	0.0729843i	\(0.0232523\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.m.b.219.5	yes	160
7.4	even	3	inner	322.2.m.b.81.4	yes	160
23.2	even	11	inner	322.2.m.b.163.4	yes	160
161.25	even	33	inner	322.2.m.b.25.5	✓	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.m.b.25.5	✓	160	161.25	even	33	inner
322.2.m.b.81.4	yes	160	7.4	even	3	inner
322.2.m.b.163.4	yes	160	23.2	even	11	inner
322.2.m.b.219.5	yes	160	1.1	even	1	trivial