Embedded newform 322.2.o.a.103.3

• Label: 322.2.o.a.103.3
• Level: $322$
• Weight: $2$
• Character: 322.103
• 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.o (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.57118294509\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(8\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			103.3
Character	\(\chi\)	\(=\)	322.103
Dual form			322.2.o.a.297.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0475819 - 0.998867i) q^{2} +(-0.673620 - 1.68262i) q^{3} +(-0.995472 + 0.0950560i) q^{4} +(0.997360 - 4.11117i) q^{5} +(-1.64866 + 0.752920i) q^{6} +(2.64472 - 0.0740405i) q^{7} +(0.142315 + 0.989821i) q^{8} +(-0.206250 + 0.196659i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 6 q^{3} + 8 q^{4} - 11 q^{7} + 16 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{9}{22}\right)\)

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
							\(3\)	−0.673620	−	1.68262i	−0.388915	−	0.971462i	−0.985616	−	0.168999i	\(-0.945947\pi\)
0.596701	−	0.802463i	\(-0.296478\pi\)
\(5\)	0.997360	−	4.11117i	0.446033	−	1.83857i	−0.0921510	−	0.995745i	\(-0.529374\pi\)
							0.538184	−	0.842827i	\(-0.319111\pi\)
\(7\)	2.64472	−	0.0740405i	0.999608	−	0.0279847i
							\(11\)	−1.93627	−	0.0922361i	−0.583809	−	0.0278102i	−0.246399	−	0.969168i	\(-0.579248\pi\)
−0.337409	+	0.941358i	\(0.609551\pi\)
\(13\)	2.09776	+	1.81772i	0.581815	+	0.504145i	0.895309	−	0.445445i	\(-0.146954\pi\)
							−0.313495	+	0.949590i	\(0.601500\pi\)
\(17\)	−0.0427760	+	0.0600705i	−0.0103747	+	0.0145692i	−0.819731	−	0.572749i	\(-0.805877\pi\)
							0.809356	+	0.587318i	\(0.199816\pi\)
\(19\)	4.27609	+	6.00493i	0.981003	+	1.37763i	0.925324	+	0.379178i	\(0.123793\pi\)
							0.0556794	+	0.998449i	\(0.482268\pi\)
\(23\)	−4.67706	+	1.06071i	−0.975234	+	0.221174i
							\(29\)	0.634173	+	1.38865i	0.117763	+	0.257865i	0.959330	−	0.282288i	\(-0.0910935\pi\)
−0.841567	+	0.540153i	\(0.818366\pi\)
\(31\)	4.70464	+	5.98244i	0.844978	+	1.07448i	0.996296	+	0.0859919i	\(0.0274059\pi\)
							−0.151318	+	0.988485i	\(0.548352\pi\)
\(37\)	−0.969908	−	1.01721i	−0.159452	−	0.167228i	0.639162	−	0.769072i	\(-0.279281\pi\)
							−0.798614	+	0.601844i	\(0.794433\pi\)
\(41\)	0.280978	+	0.956924i	0.0438814	+	0.149446i	0.978520	−	0.206154i	\(-0.0660947\pi\)
							−0.934638	+	0.355600i	\(0.884277\pi\)
\(43\)	11.5127	+	1.65527i	1.75567	+	0.252427i	0.943589	−	0.331120i	\(-0.107426\pi\)
							0.812079	+	0.583547i	\(0.198336\pi\)
\(47\)	−5.09674	−	2.94260i	−0.743436	−	0.429223i	0.0798812	−	0.996804i	\(-0.474546\pi\)
							−0.823317	+	0.567581i	\(0.807879\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.a.103.3	✓	160
7.3	odd	6	inner	322.2.o.a.241.6	yes	160
23.21	odd	22	inner	322.2.o.a.159.6	yes	160
161.136	even	66	inner	322.2.o.a.297.3	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.a.103.3	✓	160	1.1	even	1	trivial
322.2.o.a.159.6	yes	160	23.21	odd	22	inner
322.2.o.a.241.6	yes	160	7.3	odd	6	inner
322.2.o.a.297.3	yes	160	161.136	even	66	inner