Embedded newform 322.2.m.a.179.4

• Label: 322.2.m.a.179.4
• Level: $322$
• Weight: $2$
• Character: 322.179
• 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			179.4
Character	\(\chi\)	\(=\)	322.179
Dual form			322.2.m.a.9.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.235759 - 0.971812i) q^{2} +(-0.245677 - 0.709838i) q^{3} +(-0.888835 + 0.458227i) q^{4} +(1.32131 - 0.528973i) q^{5} +(-0.631908 + 0.406103i) q^{6} +(-0.478358 + 2.60215i) q^{7} +(0.654861 + 0.755750i) q^{8} +(1.91465 - 1.50569i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 4 q^{6} + 15 q^{7} + 16 q^{8} + 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{6}{11}\right)\)

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
							\(3\)	−0.245677	−	0.709838i	−0.141842	−	0.409825i	0.851596	−	0.524199i	\(-0.175635\pi\)
−0.993438	+	0.114374i	\(0.963514\pi\)
\(5\)	1.32131	−	0.528973i	0.590908	−	0.236564i	−0.0568846	−	0.998381i	\(-0.518117\pi\)
							0.647792	+	0.761817i	\(0.275692\pi\)
\(7\)	−0.478358	+	2.60215i	−0.180802	+	0.983519i
							\(11\)	1.36616	−	5.63139i	0.411913	−	1.69793i	−0.266990	−	0.963699i	\(-0.586029\pi\)
0.678903	−	0.734228i	\(-0.262456\pi\)
\(13\)	1.01087	−	2.21350i	0.280365	−	0.613914i	−0.716093	−	0.698005i	\(-0.754071\pi\)
							0.996458	+	0.0840907i	\(0.0267985\pi\)
\(17\)	−0.223279	−	4.68721i	−0.0541531	−	1.13681i	−0.848447	−	0.529281i	\(-0.822462\pi\)
							0.794294	−	0.607534i	\(-0.207841\pi\)
\(19\)	−0.281871	+	5.91719i	−0.0646655	+	1.35750i	0.700495	+	0.713657i	\(0.252963\pi\)
							−0.765161	+	0.643839i	\(0.777340\pi\)
\(23\)	−4.79393	−	0.135041i	−0.999603	−	0.0281579i
							\(29\)	0.520231	−	0.334332i	0.0966044	−	0.0620839i	−0.491447	−	0.870908i	\(-0.663532\pi\)
0.588051	+	0.808824i	\(0.299895\pi\)
\(31\)	7.00125	−	1.34938i	1.25746	−	0.242356i	0.483362	−	0.875421i	\(-0.339415\pi\)
							0.774099	+	0.633065i	\(0.218203\pi\)
\(37\)	6.14169	−	4.82988i	1.00969	−	0.794028i	0.0310293	−	0.999518i	\(-0.490121\pi\)
							0.978659	+	0.205491i	\(0.0658790\pi\)
\(41\)	−0.157842	+	1.09782i	−0.0246508	+	0.171450i	−0.998428	−	0.0560550i	\(-0.982148\pi\)
							0.973777	+	0.227505i	\(0.0730569\pi\)
\(43\)	0.408952	−	0.471956i	0.0623646	−	0.0719726i	−0.723710	−	0.690104i	\(-0.757565\pi\)
							0.786075	+	0.618132i	\(0.212110\pi\)
\(47\)	2.90919	+	5.03887i	0.424350	+	0.734995i	0.996359	−	0.0852518i	\(-0.0271695\pi\)
							−0.572010	+	0.820247i	\(0.693836\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.m.a.179.4	yes	160
7.2	even	3	inner	322.2.m.a.317.5	yes	160
23.9	even	11	inner	322.2.m.a.193.5	yes	160
161.9	even	33	inner	322.2.m.a.9.4	✓	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.m.a.9.4	✓	160	161.9	even	33	inner
322.2.m.a.179.4	yes	160	1.1	even	1	trivial
322.2.m.a.193.5	yes	160	23.9	even	11	inner
322.2.m.a.317.5	yes	160	7.2	even	3	inner