Embedded newform 322.2.m.b.193.4

• Label: 322.2.m.b.193.4
• 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			193.4
Character	\(\chi\)	\(=\)	322.193
Dual form			322.2.m.b.317.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.723734 + 0.690079i) q^{2} +(-0.170186 + 0.0328007i) q^{3} +(0.0475819 + 0.998867i) q^{4} +(-1.62995 + 1.28180i) q^{5} +(-0.145804 - 0.0937029i) q^{6} +(-2.60114 + 0.483792i) q^{7} +(-0.654861 + 0.755750i) q^{8} +(-2.75722 + 1.10382i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{2} - 2 q^{3} + 8 q^{4} + 2 q^{5} + 4 q^{6} - 11 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.170186	+	0.0328007i	−0.0982570	+	0.0189375i	−0.238143	−	0.971230i	\(-0.576539\pi\)
0.139886	+	0.990168i	\(0.455326\pi\)
\(5\)	−1.62995	+	1.28180i	−0.728935	+	0.573240i	−0.912162	−	0.409830i	\(-0.865588\pi\)
							0.183227	+	0.983071i	\(0.441346\pi\)
\(7\)	−2.60114	+	0.483792i	−0.983140	+	0.182856i
							\(11\)	−1.69378	+	1.61502i	−0.510695	+	0.486946i	−0.901048	−	0.433719i	\(-0.857201\pi\)
0.390354	+	0.920665i	\(0.372353\pi\)
\(13\)	0.0895446	+	0.196075i	0.0248352	+	0.0543815i	0.921643	−	0.388039i	\(-0.126847\pi\)
							−0.896808	+	0.442420i	\(0.854120\pi\)
\(17\)	4.71608	−	2.43131i	1.14382	−	0.589679i	0.221136	−	0.975243i	\(-0.429024\pi\)
							0.922681	+	0.385565i	\(0.125993\pi\)
\(19\)	5.87575	+	3.02916i	1.34799	+	0.694937i	0.972721	−	0.231980i	\(-0.0745203\pi\)
							0.375268	+	0.926916i	\(0.377551\pi\)
\(23\)	0.248152	+	4.78941i	0.0517433	+	0.998660i
							\(29\)	−7.00314	−	4.50065i	−1.30045	−	0.835749i	−0.307190	−	0.951648i	\(-0.599389\pi\)
−0.993261	+	0.115899i	\(0.963025\pi\)
\(31\)	0.0292696	+	0.0845690i	0.00525698	+	0.0151890i	0.947599	−	0.319463i	\(-0.103502\pi\)
							−0.942342	+	0.334652i	\(0.891381\pi\)
\(37\)	1.46977	−	0.588409i	0.241629	−	0.0967338i	−0.247687	−	0.968840i	\(-0.579671\pi\)
							0.489317	+	0.872106i	\(0.337246\pi\)
\(41\)	0.709667	+	4.93584i	0.110831	+	0.770849i	0.967114	+	0.254342i	\(0.0818588\pi\)
							−0.856283	+	0.516507i	\(0.827232\pi\)
\(43\)	2.64463	+	3.05206i	0.403302	+	0.465435i	0.920678	−	0.390323i	\(-0.127637\pi\)
							−0.517376	+	0.855758i	\(0.673091\pi\)
\(47\)	−1.08642	−	1.88173i	−0.158470	−	0.274479i	0.775847	−	0.630921i	\(-0.217323\pi\)
							−0.934317	+	0.356443i	\(0.883990\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.m.b.193.4	yes	160
7.2	even	3	inner	322.2.m.b.9.5	✓	160
23.18	even	11	inner	322.2.m.b.179.5	yes	160
161.156	even	33	inner	322.2.m.b.317.4	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.m.b.9.5	✓	160	7.2	even	3	inner
322.2.m.b.179.5	yes	160	23.18	even	11	inner
322.2.m.b.193.4	yes	160	1.1	even	1	trivial
322.2.m.b.317.4	yes	160	161.156	even	33	inner