Embedded newform 322.2.m.a.81.1

• Label: 322.2.m.a.81.1
• Level: $322$
• Weight: $2$
• Character: 322.81
• 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			81.1
Character	\(\chi\)	\(=\)	322.81
Dual form			322.2.m.a.163.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.888835 + 0.458227i) q^{2} +(-2.06678 - 1.62534i) q^{3} +(0.580057 + 0.814576i) q^{4} +(2.01734 + 1.92353i) q^{5} +(-1.09226 - 2.39171i) q^{6} +(-1.50150 + 2.17842i) q^{7} +(0.142315 + 0.989821i) q^{8} +(0.922599 + 3.80300i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 4 q^{6} + 15 q^{7} + 16 q^{8} + 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{10}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.888835	+	0.458227i	0.628502	+	0.324015i
							\(3\)	−2.06678	−	1.62534i	−1.19326	−	0.938389i	−0.194165	−	0.980969i	\(-0.562200\pi\)
−0.999093	+	0.0425804i	\(0.986442\pi\)
\(5\)	2.01734	+	1.92353i	0.902184	+	0.860230i	0.990731	−	0.135842i	\(-0.0433739\pi\)
							−0.0885470	+	0.996072i	\(0.528222\pi\)
\(7\)	−1.50150	+	2.17842i	−0.567514	+	0.823364i
							\(11\)	5.31775	−	2.74149i	1.60336	−	0.826591i	0.603797	−	0.797138i	\(-0.293654\pi\)
0.999566	−	0.0294529i	\(-0.00937652\pi\)
\(13\)	−4.01819	+	4.63724i	−1.11445	+	1.28614i	−0.160211	+	0.987083i	\(0.551218\pi\)
							−0.954235	+	0.299057i	\(0.903328\pi\)
\(17\)	5.04251	+	0.481501i	1.22299	+	0.116781i	0.686502	−	0.727128i	\(-0.259145\pi\)
							0.536486	+	0.843909i	\(0.319751\pi\)
\(19\)	2.11239	−	0.201709i	0.484616	−	0.0462752i	0.150111	−	0.988669i	\(-0.452037\pi\)
							0.334505	+	0.942394i	\(0.391431\pi\)
\(23\)	4.54501	+	1.53065i	0.947700	+	0.319163i
							\(29\)	0.0745364	+	0.163212i	0.0138411	+	0.0303077i	0.916426	−	0.400204i	\(-0.131061\pi\)
−0.902585	+	0.430511i	\(0.858333\pi\)
\(31\)	−6.19979	−	2.48202i	−1.11352	−	0.445784i	−0.259349	−	0.965784i	\(-0.583508\pi\)
							−0.854167	+	0.519999i	\(0.825932\pi\)
\(37\)	0.560991	+	2.31243i	0.0922263	+	0.380162i	0.999154	−	0.0411243i	\(-0.0130940\pi\)
							−0.906928	+	0.421286i	\(0.861579\pi\)
\(41\)	−3.49179	+	1.02528i	−0.545326	+	0.160122i	−0.542780	−	0.839875i	\(-0.682628\pi\)
							−0.00254543	+	0.999997i	\(0.500810\pi\)
\(43\)	0.637473	−	4.43373i	0.0972138	−	0.676137i	−0.881692	−	0.471825i	\(-0.843595\pi\)
							0.978906	−	0.204312i	\(-0.0654956\pi\)
\(47\)	−5.59447	−	9.68991i	−0.816038	−	1.41342i	−0.908580	−	0.417710i	\(-0.862833\pi\)
							0.0925423	−	0.995709i	\(-0.470501\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.m.a.81.1	yes	160
7.2	even	3	inner	322.2.m.a.219.8	yes	160
23.2	even	11	inner	322.2.m.a.25.8	✓	160
161.2	even	33	inner	322.2.m.a.163.1	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.m.a.25.8	✓	160	23.2	even	11	inner
322.2.m.a.81.1	yes	160	1.1	even	1	trivial
322.2.m.a.163.1	yes	160	161.2	even	33	inner
322.2.m.a.219.8	yes	160	7.2	even	3	inner