Embedded newform 867.2.i.e.65.4

• Label: 867.2.i.e.65.4
• Level: $867$
• Weight: $2$
• Character: 867.65
• Analytic conductor: $6.923$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $32$

Newspace parameters

Level:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	867.i (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.92302985525\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(4\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			65.4
Character	\(\chi\)	\(=\)	867.65
Dual form			867.2.i.e.827.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.855706 + 2.06586i) q^{2} +(0.385055 + 1.68871i) q^{3} +(-2.12132 + 2.12132i) q^{4} +(-1.57139 - 2.35175i) q^{5} +(-3.15913 + 2.24051i) q^{6} +(-2.62934 - 1.75687i) q^{7} +(-2.06586 - 0.855706i) q^{8} +(-2.70347 + 1.30049i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/867\mathbb{Z}\right)^\times\).

\(n\)	\(290\)	\(292\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.855706	+	2.06586i	0.605076	+	1.46078i	0.868296	+	0.496046i	\(0.165215\pi\)
							−0.263221	+	0.964736i	\(0.584785\pi\)
\(3\)	0.385055	+	1.68871i	0.222312	+	0.974976i
							\(5\)	−1.57139	−	2.35175i	−0.702747	−	1.05174i	−0.995427	−	0.0955247i	\(-0.969547\pi\)
0.292680	−	0.956210i	\(-0.405453\pi\)
\(7\)	−2.62934	−	1.75687i	−0.993796	−	0.664033i	−0.0514510	−	0.998676i	\(-0.516385\pi\)
							−0.942345	+	0.334642i	\(0.891385\pi\)
\(11\)	−0.275899	−	1.38704i	−0.0831868	−	0.418208i	−0.999829	−	0.0184992i	\(-0.994111\pi\)
							0.916642	−	0.399709i	\(-0.130889\pi\)
\(13\)	−2.82843	−	2.82843i	−0.784465	−	0.784465i	0.196116	−	0.980581i	\(-0.437167\pi\)
							−0.980581	+	0.196116i	\(0.937167\pi\)
\(17\)		0			0
							\(19\)		0			0		−0.382683	−	0.923880i	\(-0.625000\pi\)
0.382683	+	0.923880i	\(0.375000\pi\)
\(23\)	−6.93520	+	1.37950i	−1.44609	+	0.287645i	−0.854860	−	0.518859i	\(-0.826357\pi\)
							−0.591229	+	0.806504i	\(0.701357\pi\)
\(29\)	2.35175	−	1.57139i	0.436709	−	0.291800i	−0.317716	−	0.948186i	\(-0.602916\pi\)
							0.754425	+	0.656386i	\(0.227916\pi\)
\(31\)	3.10152	+	0.616930i	0.557049	+	0.110804i	0.465583	−	0.885004i	\(-0.345845\pi\)
							0.0914659	+	0.995808i	\(0.470845\pi\)
\(37\)	1.23386	−	6.20303i	0.202845	−	1.01977i	−0.736406	−	0.676540i	\(-0.763478\pi\)
							0.939251	−	0.343232i	\(-0.111522\pi\)
\(41\)	−3.14278	+	4.70350i	−0.490820	+	0.734564i	−0.991363	−	0.131148i	\(-0.958134\pi\)
							0.500543	+	0.865712i	\(0.333134\pi\)
\(43\)	3.69552	+	1.53073i	0.563561	+	0.233435i	0.646230	−	0.763142i	\(-0.276344\pi\)
							−0.0826692	+	0.996577i	\(0.526344\pi\)
\(47\)	−6.32456	+	6.32456i	−0.922531	+	0.922531i	−0.997208	−	0.0746766i	\(-0.976208\pi\)
							0.0746766	+	0.997208i	\(0.476208\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.2.i.e.65.4	yes	32
3.2	odd	2	inner	867.2.i.e.65.1	✓	32
17.2	even	8	inner	867.2.i.e.158.2	yes	32
17.3	odd	16	inner	867.2.i.e.653.3	yes	32
17.4	even	4	inner	867.2.i.e.224.2	yes	32
17.5	odd	16	inner	867.2.i.e.503.1	yes	32
17.6	odd	16	inner	867.2.i.e.827.2	yes	32
17.7	odd	16	inner	867.2.i.e.329.4	yes	32
17.8	even	8	inner	867.2.i.e.131.4	yes	32
17.9	even	8	inner	867.2.i.e.131.3	yes	32
17.10	odd	16	inner	867.2.i.e.329.3	yes	32
17.11	odd	16	inner	867.2.i.e.827.1	yes	32
17.12	odd	16	inner	867.2.i.e.503.2	yes	32
17.13	even	4	inner	867.2.i.e.224.1	yes	32
17.14	odd	16	inner	867.2.i.e.653.4	yes	32
17.15	even	8	inner	867.2.i.e.158.1	yes	32
17.16	even	2	inner	867.2.i.e.65.3	yes	32
51.2	odd	8	inner	867.2.i.e.158.4	yes	32
51.5	even	16	inner	867.2.i.e.503.3	yes	32
51.8	odd	8	inner	867.2.i.e.131.2	yes	32
51.11	even	16	inner	867.2.i.e.827.4	yes	32
51.14	even	16	inner	867.2.i.e.653.2	yes	32
51.20	even	16	inner	867.2.i.e.653.1	yes	32
51.23	even	16	inner	867.2.i.e.827.3	yes	32
51.26	odd	8	inner	867.2.i.e.131.1	yes	32
51.29	even	16	inner	867.2.i.e.503.4	yes	32
51.32	odd	8	inner	867.2.i.e.158.3	yes	32
51.38	odd	4	inner	867.2.i.e.224.4	yes	32
51.41	even	16	inner	867.2.i.e.329.2	yes	32
51.44	even	16	inner	867.2.i.e.329.1	yes	32
51.47	odd	4	inner	867.2.i.e.224.3	yes	32
51.50	odd	2	inner	867.2.i.e.65.2	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
867.2.i.e.65.1	✓	32	3.2	odd	2	inner
867.2.i.e.65.2	yes	32	51.50	odd	2	inner
867.2.i.e.65.3	yes	32	17.16	even	2	inner
867.2.i.e.65.4	yes	32	1.1	even	1	trivial
867.2.i.e.131.1	yes	32	51.26	odd	8	inner
867.2.i.e.131.2	yes	32	51.8	odd	8	inner
867.2.i.e.131.3	yes	32	17.9	even	8	inner
867.2.i.e.131.4	yes	32	17.8	even	8	inner
867.2.i.e.158.1	yes	32	17.15	even	8	inner
867.2.i.e.158.2	yes	32	17.2	even	8	inner
867.2.i.e.158.3	yes	32	51.32	odd	8	inner
867.2.i.e.158.4	yes	32	51.2	odd	8	inner
867.2.i.e.224.1	yes	32	17.13	even	4	inner
867.2.i.e.224.2	yes	32	17.4	even	4	inner
867.2.i.e.224.3	yes	32	51.47	odd	4	inner
867.2.i.e.224.4	yes	32	51.38	odd	4	inner
867.2.i.e.329.1	yes	32	51.44	even	16	inner
867.2.i.e.329.2	yes	32	51.41	even	16	inner
867.2.i.e.329.3	yes	32	17.10	odd	16	inner
867.2.i.e.329.4	yes	32	17.7	odd	16	inner
867.2.i.e.503.1	yes	32	17.5	odd	16	inner
867.2.i.e.503.2	yes	32	17.12	odd	16	inner
867.2.i.e.503.3	yes	32	51.5	even	16	inner
867.2.i.e.503.4	yes	32	51.29	even	16	inner
867.2.i.e.653.1	yes	32	51.20	even	16	inner
867.2.i.e.653.2	yes	32	51.14	even	16	inner
867.2.i.e.653.3	yes	32	17.3	odd	16	inner
867.2.i.e.653.4	yes	32	17.14	odd	16	inner
867.2.i.e.827.1	yes	32	17.11	odd	16	inner
867.2.i.e.827.2	yes	32	17.6	odd	16	inner
867.2.i.e.827.3	yes	32	51.23	even	16	inner
867.2.i.e.827.4	yes	32	51.11	even	16	inner