Embedded newform 322.2.m.b.193.7

• Label: 322.2.m.b.193.7
• 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			193.7
Character	\(\chi\)	\(=\)	322.193
Dual form			322.2.m.b.317.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.723734 + 0.690079i) q^{2} +(2.18803 - 0.421708i) q^{3} +(0.0475819 + 0.998867i) q^{4} +(-0.0648833 + 0.0510248i) q^{5} +(1.87456 + 1.20471i) q^{6} +(-0.143694 - 2.64185i) q^{7} +(-0.654861 + 0.755750i) q^{8} +(1.82452 - 0.730427i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{2} - 2 q^{3} + 8 q^{4} + 2 q^{5} + 4 q^{6} - 11 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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
							\(3\)	2.18803	−	0.421708i	1.26326	−	0.243473i	0.486728	−	0.873554i	\(-0.338190\pi\)
0.776530	+	0.630081i	\(0.216978\pi\)
\(5\)	−0.0648833	+	0.0510248i	−0.0290167	+	0.0228190i	−0.632562	−	0.774510i	\(-0.717997\pi\)
							0.603545	+	0.797329i	\(0.293754\pi\)
\(7\)	−0.143694	−	2.64185i	−0.0543113	−	0.998524i
							\(11\)	1.64691	−	1.57032i	0.496562	−	0.473470i	−0.399921	−	0.916550i	\(-0.630962\pi\)
0.896482	+	0.443079i	\(0.146114\pi\)
\(13\)	2.29274	+	5.02040i	0.635892	+	1.39241i	0.903377	+	0.428847i	\(0.141080\pi\)
							−0.267485	+	0.963562i	\(0.586193\pi\)
\(17\)	−0.108352	+	0.0558591i	−0.0262791	+	0.0135478i	−0.471316	−	0.881964i	\(-0.656221\pi\)
							0.445037	+	0.895512i	\(0.353191\pi\)
\(19\)	−4.83613	−	2.49320i	−1.10949	−	0.571979i	−0.196718	−	0.980460i	\(-0.563028\pi\)
							−0.912767	+	0.408481i	\(0.866059\pi\)
\(23\)	−2.44448	+	4.12608i	−0.509709	+	0.860347i
							\(29\)	−1.95651	−	1.25737i	−0.363315	−	0.233488i	0.346228	−	0.938150i	\(-0.387462\pi\)
−0.709543	+	0.704662i	\(0.751099\pi\)
\(31\)	−3.10166	−	8.96165i	−0.557074	−	1.60956i	−0.773871	−	0.633343i	\(-0.781682\pi\)
							0.216797	−	0.976217i	\(-0.430439\pi\)
\(37\)	1.23964	−	0.496278i	0.203796	−	0.0815876i	−0.267519	−	0.963553i	\(-0.586204\pi\)
							0.471315	+	0.881965i	\(0.343780\pi\)
\(41\)	−1.07590	−	7.48304i	−0.168027	−	1.16865i	−0.882955	−	0.469457i	\(-0.844450\pi\)
							0.714928	−	0.699198i	\(-0.246459\pi\)
\(43\)	−3.73669	−	4.31237i	−0.569839	−	0.657630i	0.395550	−	0.918445i	\(-0.370554\pi\)
							−0.965389	+	0.260815i	\(0.916009\pi\)
\(47\)	4.11198	+	7.12215i	0.599793	+	1.03887i	0.992851	+	0.119359i	\(0.0380838\pi\)
							−0.393058	+	0.919514i	\(0.628583\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.m.b.193.7	yes	160
7.2	even	3	inner	322.2.m.b.9.2	✓	160
23.18	even	11	inner	322.2.m.b.179.2	yes	160
161.156	even	33	inner	322.2.m.b.317.7	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.m.b.9.2	✓	160	7.2	even	3	inner
322.2.m.b.179.2	yes	160	23.18	even	11	inner
322.2.m.b.193.7	yes	160	1.1	even	1	trivial
322.2.m.b.317.7	yes	160	161.156	even	33	inner