Embedded newform 322.2.m.a.193.7

• Label: 322.2.m.a.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.a.317.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.723734 - 0.690079i) q^{2} +(1.37247 - 0.264521i) q^{3} +(0.0475819 + 0.998867i) q^{4} +(3.39605 - 2.67068i) q^{5} +(-1.17584 - 0.755667i) q^{6} +(0.480218 - 2.60181i) q^{7} +(0.654861 - 0.755750i) q^{8} +(-0.971411 + 0.388894i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 4 q^{6} + 15 q^{7} + 16 q^{8} + 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{11}\right)\)

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
							\(3\)	1.37247	−	0.264521i	0.792394	−	0.152721i	0.223035	−	0.974810i	\(-0.428404\pi\)
0.569359	+	0.822089i	\(0.307191\pi\)
\(5\)	3.39605	−	2.67068i	1.51876	−	1.19437i	0.599714	−	0.800214i	\(-0.295281\pi\)
							0.919045	−	0.394151i	\(-0.128961\pi\)
\(7\)	0.480218	−	2.60181i	0.181505	−	0.983390i
							\(11\)	−1.31532	+	1.25416i	−0.396585	+	0.378143i	−0.861965	−	0.506968i	\(-0.830766\pi\)
0.465380	+	0.885111i	\(0.345918\pi\)
\(13\)	0.729259	+	1.59685i	0.202260	+	0.442888i	0.983396	−	0.181473i	\(-0.0580866\pi\)
							−0.781136	+	0.624361i	\(0.785359\pi\)
\(17\)	−6.78275	+	3.49675i	−1.64506	+	0.848086i	−0.648865	+	0.760903i	\(0.724756\pi\)
							−0.996192	+	0.0871826i	\(0.972214\pi\)
\(19\)	2.33203	+	1.20225i	0.535005	+	0.275814i	0.704484	−	0.709720i	\(-0.251178\pi\)
							−0.169479	+	0.985534i	\(0.554209\pi\)
\(23\)	4.26633	+	2.19054i	0.889591	+	0.456758i
							\(29\)	−1.63930	−	1.05352i	−0.304411	−	0.195633i	0.379513	−	0.925186i	\(-0.376092\pi\)
−0.683924	+	0.729553i	\(0.739728\pi\)
\(31\)	1.49204	+	4.31098i	0.267979	+	0.774274i	0.996052	+	0.0887745i	\(0.0282950\pi\)
							−0.728073	+	0.685500i	\(0.759584\pi\)
\(37\)	3.29794	−	1.32029i	0.542177	−	0.217055i	−0.0843754	−	0.996434i	\(-0.526890\pi\)
							0.626553	+	0.779379i	\(0.284465\pi\)
\(41\)	−0.955048	−	6.64250i	−0.149153	−	1.03738i	−0.917610	−	0.397483i	\(-0.869884\pi\)
							0.768456	−	0.639902i	\(-0.221025\pi\)
\(43\)	3.75086	+	4.32872i	0.572000	+	0.660124i	0.965865	−	0.259045i	\(-0.0834077\pi\)
							−0.393865	+	0.919168i	\(0.628862\pi\)
\(47\)	0.638927	+	1.10665i	0.0931971	+	0.161422i	0.908855	−	0.417113i	\(-0.136958\pi\)
							−0.815658	+	0.578535i	\(0.803625\pi\)

(See \(a_n\) instead)

Twists

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

    

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