Embedded newform 322.2.m.a.9.1

• Label: 322.2.m.a.9.1
• Level: $322$
• Weight: $2$
• Character: 322.9
• 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			9.1
Character	\(\chi\)	\(=\)	322.9
Dual form			322.2.m.a.179.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.235759 + 0.971812i) q^{2} +(-0.899762 + 2.59969i) q^{3} +(-0.888835 - 0.458227i) q^{4} +(-0.684642 - 0.274089i) q^{5} +(-2.31428 - 1.48730i) q^{6} +(-2.64563 - 0.0257032i) q^{7} +(0.654861 - 0.755750i) q^{8} +(-3.59067 - 2.82373i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 4 q^{6} + 15 q^{7} + 16 q^{8} + 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.235759	+	0.971812i	−0.166707	+	0.687175i
							\(3\)	−0.899762	+	2.59969i	−0.519478	+	1.50093i	0.314566	+	0.949235i	\(0.398141\pi\)
−0.834044	+	0.551697i	\(0.813980\pi\)
\(5\)	−0.684642	−	0.274089i	−0.306181	−	0.122577i	0.213484	−	0.976947i	\(-0.431519\pi\)
							−0.519665	+	0.854370i	\(0.673943\pi\)
\(7\)	−2.64563	−	0.0257032i	−0.999953	−	0.00971488i
							\(11\)	−0.664233	−	2.73801i	−0.200274	−	0.825540i	−0.979932	−	0.199334i	\(-0.936122\pi\)
0.779658	−	0.626206i	\(-0.215393\pi\)
\(13\)	−0.159553	−	0.349373i	−0.0442521	−	0.0968987i	0.886215	−	0.463275i	\(-0.153326\pi\)
							−0.930467	+	0.366376i	\(0.880598\pi\)
\(17\)	−0.131212	+	2.75447i	−0.0318235	+	0.668057i	0.925449	+	0.378871i	\(0.123688\pi\)
							−0.957273	+	0.289186i	\(0.906615\pi\)
\(19\)	0.103389	+	2.17039i	0.0237190	+	0.497922i	0.979203	+	0.202881i	\(0.0650304\pi\)
							−0.955484	+	0.295041i	\(0.904667\pi\)
\(23\)	−0.449854	+	4.77469i	−0.0938010	+	0.995591i
							\(29\)	−5.10237	−	3.27909i	−0.947486	−	0.608912i	−0.0269779	−	0.999636i	\(-0.508588\pi\)
−0.920508	+	0.390724i	\(0.872225\pi\)
\(31\)	1.18628	+	0.228637i	0.213063	+	0.0410645i	0.294665	−	0.955600i	\(-0.404792\pi\)
							−0.0816028	+	0.996665i	\(0.526004\pi\)
\(37\)	−2.50137	−	1.96710i	−0.411222	−	0.323389i	0.391026	−	0.920380i	\(-0.372120\pi\)
							−0.802248	+	0.596991i	\(0.796363\pi\)
\(41\)	−1.55393	−	10.8078i	−0.242683	−	1.68790i	−0.638542	−	0.769587i	\(-0.720462\pi\)
							0.395859	−	0.918311i	\(-0.370447\pi\)
\(43\)	1.27797	+	1.47486i	0.194889	+	0.224914i	0.844780	−	0.535113i	\(-0.179731\pi\)
							−0.649891	+	0.760027i	\(0.725186\pi\)
\(47\)	−4.45492	+	7.71615i	−0.649817	+	1.12552i	0.333349	+	0.942803i	\(0.391821\pi\)
							−0.983166	+	0.182713i	\(0.941512\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.m.a.9.1	✓	160
7.4	even	3	inner	322.2.m.a.193.8	yes	160
23.18	even	11	inner	322.2.m.a.317.8	yes	160
161.18	even	33	inner	322.2.m.a.179.1	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.m.a.9.1	✓	160	1.1	even	1	trivial
322.2.m.a.179.1	yes	160	161.18	even	33	inner
322.2.m.a.193.8	yes	160	7.4	even	3	inner
322.2.m.a.317.8	yes	160	23.18	even	11	inner