Embedded newform 322.2.m.a.25.1

• Label: 322.2.m.a.25.1
• Level: $322$
• Weight: $2$
• Character: 322.25
• 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			25.1
Character	\(\chi\)	\(=\)	322.25
Dual form			322.2.m.a.219.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0475819 + 0.998867i) q^{2} +(-3.01871 - 1.20851i) q^{3} +(-0.995472 - 0.0950560i) q^{4} +(-0.530176 - 2.18541i) q^{5} +(1.35077 - 2.95778i) q^{6} +(-2.53173 - 0.768337i) q^{7} +(0.142315 - 0.989821i) q^{8} +(5.48089 + 5.22602i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 4 q^{6} + 15 q^{7} + 16 q^{8} + 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{11}\right)\)

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
							\(3\)	−3.01871	−	1.20851i	−1.74285	−	0.697732i	−0.999587	−	0.0287328i	\(-0.990853\pi\)
−0.743263	−	0.668999i	\(-0.766723\pi\)
\(5\)	−0.530176	−	2.18541i	−0.237102	−	0.977347i	−0.958560	−	0.284889i	\(-0.908043\pi\)
							0.721459	−	0.692458i	\(-0.243472\pi\)
\(7\)	−2.53173	−	0.768337i	−0.956904	−	0.290404i
							\(11\)	0.185687	+	3.89806i	0.0559868	+	1.17531i	0.835705	+	0.549178i	\(0.185059\pi\)
−0.779718	+	0.626130i	\(0.784638\pi\)
\(13\)	1.52164	+	1.75606i	0.422026	+	0.487044i	0.926453	−	0.376411i	\(-0.122842\pi\)
							−0.504427	+	0.863454i	\(0.668296\pi\)
\(17\)	0.540778	+	0.759417i	0.131158	+	0.184186i	0.874932	−	0.484245i	\(-0.160906\pi\)
							−0.743774	+	0.668431i	\(0.766966\pi\)
\(19\)	−1.39297	+	1.95615i	−0.319569	+	0.448773i	−0.942846	−	0.333229i	\(-0.891862\pi\)
							0.623276	+	0.782002i	\(0.285801\pi\)
\(23\)	1.75259	−	4.46413i	0.365439	−	0.930835i
							\(29\)	−4.31224	+	9.44249i	−0.800763	+	1.75343i	−0.157886	+	0.987457i	\(0.550468\pi\)
−0.642877	+	0.765969i	\(0.722259\pi\)
\(31\)	0.707584	+	0.556450i	0.127086	+	0.0999415i	0.679679	−	0.733510i	\(-0.262119\pi\)
							−0.552593	+	0.833451i	\(0.686362\pi\)
\(37\)	2.08153	+	1.98474i	0.342202	+	0.326289i	0.841618	−	0.540073i	\(-0.181603\pi\)
							−0.499416	+	0.866362i	\(0.666452\pi\)
\(41\)	−0.706645	−	0.207490i	−0.110359	−	0.0324044i	0.226087	−	0.974107i	\(-0.427407\pi\)
							−0.336446	+	0.941703i	\(0.609225\pi\)
\(43\)	1.08968	+	7.57887i	0.166174	+	1.15577i	0.886702	+	0.462341i	\(0.152990\pi\)
							−0.720528	+	0.693426i	\(0.756101\pi\)
\(47\)	5.89679	+	10.2135i	0.860136	+	1.48980i	0.871797	+	0.489867i	\(0.162955\pi\)
							−0.0116617	+	0.999932i	\(0.503712\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.m.a.25.1	✓	160
7.2	even	3	inner	322.2.m.a.163.8	yes	160
23.12	even	11	inner	322.2.m.a.81.8	yes	160
161.58	even	33	inner	322.2.m.a.219.1	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.m.a.25.1	✓	160	1.1	even	1	trivial
322.2.m.a.81.8	yes	160	23.12	even	11	inner
322.2.m.a.163.8	yes	160	7.2	even	3	inner
322.2.m.a.219.1	yes	160	161.58	even	33	inner