Embedded newform 861.2.l.a.419.17

• Label: 861.2.l.a.419.17
• Level: $861$
• Weight: $2$
• Character: 861.419
• Analytic conductor: $6.875$
• Analytic rank: $0$
• Dimension: $216$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 861 = 3 \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	861.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.87511961403\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(108\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			419.17
Character	\(\chi\)	\(=\)	861.419
Dual form			861.2.l.a.524.17

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.06720 q^{2} +(-1.24187 - 1.20738i) q^{3} +2.27332 q^{4} -4.31335i q^{5} +(2.56719 + 2.49590i) q^{6} +(0.905579 - 2.48595i) q^{7} -0.565003 q^{8} +(0.0844609 + 2.99881i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q + 192 q^{4} - 4 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/861\mathbb{Z}\right)^\times\).

\(n\)	\(211\)	\(493\)	\(575\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.06720			−1.46173			−0.730866	−	0.682521i	\(-0.760883\pi\)
							−0.730866	+	0.682521i	\(0.760883\pi\)
\(3\)	−1.24187	−	1.20738i	−0.716992	−	0.697082i
							\(5\)		−	4.31335i		−	1.92899i	−0.264103	−	0.964494i	\(-0.585076\pi\)
0.264103	−	0.964494i	\(-0.414924\pi\)
\(7\)	0.905579	−	2.48595i	0.342277	−	0.939599i
							\(11\)	−2.49995	+	2.49995i	−0.753763	+	0.753763i	−0.975179	−	0.221416i	\(-0.928932\pi\)
0.221416	+	0.975179i	\(0.428932\pi\)
\(13\)	0.165020	−	0.165020i	0.0457684	−	0.0457684i	−0.683852	−	0.729621i	\(-0.739697\pi\)
							0.729621	+	0.683852i	\(0.239697\pi\)
\(17\)	−1.98789	+	1.98789i	−0.482135	+	0.482135i	−0.905813	−	0.423678i	\(-0.860739\pi\)
							0.423678	+	0.905813i	\(0.360739\pi\)
\(19\)	−2.08349	−	2.08349i	−0.477985	−	0.477985i	0.426502	−	0.904487i	\(-0.359746\pi\)
							−0.904487	+	0.426502i	\(0.859746\pi\)
\(23\)			4.46968i			0.931993i	0.884787	+	0.465996i	\(0.154304\pi\)
							−0.884787	+	0.465996i	\(0.845696\pi\)
\(29\)	2.03858	−	2.03858i	0.378555	−	0.378555i	−0.492026	−	0.870581i	\(-0.663743\pi\)
							0.870581	+	0.492026i	\(0.163743\pi\)
\(31\)			6.60557i			1.18640i	0.805057	+	0.593198i	\(0.202135\pi\)
							−0.805057	+	0.593198i	\(0.797865\pi\)
\(37\)	7.79826			1.28203			0.641013	−	0.767530i	\(-0.278514\pi\)
							0.641013	+	0.767530i	\(0.278514\pi\)
\(41\)	−3.31014	−	5.48115i	−0.516958	−	0.856011i
							\(43\)		−	1.54443i		−	0.235523i	−0.993042	−	0.117761i	\(-0.962428\pi\)
0.993042	−	0.117761i	\(-0.0375718\pi\)
\(47\)	−6.21619	+	6.21619i	−0.906725	+	0.906725i	−0.996006	−	0.0892815i	\(-0.971543\pi\)
							0.0892815	+	0.996006i	\(0.471543\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	861.2.l.a.419.17	✓	216
3.2	odd	2	inner	861.2.l.a.419.92	yes	216
7.6	odd	2	inner	861.2.l.a.419.18	yes	216
21.20	even	2	inner	861.2.l.a.419.91	yes	216
41.32	even	4	inner	861.2.l.a.524.91	yes	216
123.32	odd	4	inner	861.2.l.a.524.18	yes	216
287.237	odd	4	inner	861.2.l.a.524.92	yes	216
861.524	even	4	inner	861.2.l.a.524.17	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
861.2.l.a.419.17	✓	216	1.1	even	1	trivial
861.2.l.a.419.18	yes	216	7.6	odd	2	inner
861.2.l.a.419.91	yes	216	21.20	even	2	inner
861.2.l.a.419.92	yes	216	3.2	odd	2	inner
861.2.l.a.524.17	yes	216	861.524	even	4	inner
861.2.l.a.524.18	yes	216	123.32	odd	4	inner
861.2.l.a.524.91	yes	216	41.32	even	4	inner
861.2.l.a.524.92	yes	216	287.237	odd	4	inner