Embedded newform 322.2.m.a.9.3

• Label: 322.2.m.a.9.3
• 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			9.3
Character	\(\chi\)	\(=\)	322.9
Dual form			322.2.m.a.179.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.235759 + 0.971812i) q^{2} +(-0.289551 + 0.836604i) q^{3} +(-0.888835 - 0.458227i) q^{4} +(2.56790 + 1.02803i) q^{5} +(-0.744757 - 0.478626i) q^{6} +(2.64344 - 0.110492i) q^{7} +(0.654861 - 0.755750i) q^{8} +(1.74209 + 1.37000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 4 q^{6} + 15 q^{7} + 16 q^{8} + 28 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.289551	+	0.836604i	−0.167173	+	0.483013i	−0.997206	−	0.0747001i	\(-0.976200\pi\)
0.830033	+	0.557714i	\(0.188321\pi\)
\(5\)	2.56790	+	1.02803i	1.14840	+	0.459749i	0.866245	−	0.499620i	\(-0.166527\pi\)
							0.282153	+	0.959369i	\(0.408951\pi\)
\(7\)	2.64344	−	0.110492i	0.999128	−	0.0417622i
							\(11\)	−1.07352	−	4.42509i	−0.323677	−	1.33422i	−0.868510	−	0.495671i	\(-0.834922\pi\)
0.544833	−	0.838545i	\(-0.316593\pi\)
\(13\)	−0.544222	−	1.19168i	−0.150940	−	0.330512i	0.819025	−	0.573758i	\(-0.194515\pi\)
							−0.969965	+	0.243246i	\(0.921788\pi\)
\(17\)	−0.0300454	+	0.630731i	−0.00728708	+	0.152975i	0.992291	+	0.123930i	\(0.0395497\pi\)
							−0.999578	+	0.0290451i	\(0.990753\pi\)
\(19\)	0.183937	+	3.86131i	0.0421980	+	0.885845i	0.916395	+	0.400275i	\(0.131085\pi\)
							−0.874197	+	0.485571i	\(0.838612\pi\)
\(23\)	−0.531254	+	4.76632i	−0.110774	+	0.993846i
							\(29\)	2.57208	+	1.65298i	0.477623	+	0.306950i	0.757211	−	0.653170i	\(-0.226561\pi\)
−0.279588	+	0.960120i	\(0.590198\pi\)
\(31\)	−6.94398	−	1.33834i	−1.24718	−	0.240373i	−0.477399	−	0.878687i	\(-0.658420\pi\)
							−0.769777	+	0.638313i	\(0.779632\pi\)
\(37\)	−5.68268	−	4.46891i	−0.934227	−	0.734685i	0.0300585	−	0.999548i	\(-0.490431\pi\)
							−0.964286	+	0.264864i	\(0.914673\pi\)
\(41\)	0.705702	+	4.90827i	0.110212	+	0.766543i	0.967712	+	0.252058i	\(0.0811075\pi\)
							−0.857500	+	0.514484i	\(0.827983\pi\)
\(43\)	−8.40891	−	9.70440i	−1.28235	−	1.47991i	−0.794880	−	0.606767i	\(-0.792466\pi\)
							−0.487468	−	0.873141i	\(-0.662079\pi\)
\(47\)	6.14649	−	10.6460i	0.896558	−	1.55288i	0.0646930	−	0.997905i	\(-0.479393\pi\)
							0.831865	−	0.554978i	\(-0.187273\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.m.a.9.3	✓	160
7.4	even	3	inner	322.2.m.a.193.6	yes	160
23.18	even	11	inner	322.2.m.a.317.6	yes	160
161.18	even	33	inner	322.2.m.a.179.3	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.m.a.9.3	✓	160	1.1	even	1	trivial
322.2.m.a.179.3	yes	160	161.18	even	33	inner
322.2.m.a.193.6	yes	160	7.4	even	3	inner
322.2.m.a.317.6	yes	160	23.18	even	11	inner