Embedded newform 32.8.g.a.13.13

• Label: 32.8.g.a.13.13
• Level: $32$
• Weight: $8$
• Character: 32.13
• Analytic conductor: $9.996$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	32.g (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.99632081549\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(27\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			13.13
Character	\(\chi\)	\(=\)	32.13
Dual form			32.8.g.a.5.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.49503 - 11.0352i) q^{2} +(26.3289 - 63.5636i) q^{3} +(-115.550 + 55.0661i) q^{4} +(-124.019 + 51.3705i) q^{5} +(-767.126 - 131.951i) q^{6} +(-765.180 + 765.180i) q^{7} +(895.963 + 1137.72i) q^{8} +(-1800.68 - 1800.68i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{7}{8}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.49503	−	11.0352i	−0.220531	−	0.975380i
							\(3\)	26.3289	−	63.5636i	0.563000	−	1.35920i	−0.344356	−	0.938839i	\(-0.611903\pi\)
0.907356	−	0.420363i	\(-0.138097\pi\)
\(5\)	−124.019	+	51.3705i	−0.443705	+	0.183789i	−0.593339	−	0.804953i	\(-0.702191\pi\)
							0.149634	+	0.988741i	\(0.452191\pi\)
\(7\)	−765.180	+	765.180i	−0.843180	+	0.843180i	−0.989271	−	0.146091i	\(-0.953331\pi\)
							0.146091	+	0.989271i	\(0.453331\pi\)
\(11\)	−2241.66	−	5411.83i	−0.507802	−	1.22594i	−0.945146	−	0.326648i	\(-0.894081\pi\)
							0.437344	−	0.899294i	\(-0.355919\pi\)
\(13\)	3368.32	+	1395.20i	0.425217	+	0.176131i	0.585021	−	0.811018i	\(-0.301086\pi\)
							−0.159804	+	0.987149i	\(0.551086\pi\)
\(17\)			23848.3i			1.17730i	0.808390	+	0.588648i	\(0.200340\pi\)
							−0.808390	+	0.588648i	\(0.799660\pi\)
\(19\)	−41908.6	−	17359.1i	−1.40173	−	0.580617i	−0.451532	−	0.892255i	\(-0.649122\pi\)
							−0.950201	+	0.311638i	\(0.899122\pi\)
\(23\)	−5869.45	−	5869.45i	−0.100589	−	0.100589i	0.655021	−	0.755610i	\(-0.272660\pi\)
							−0.755610	+	0.655021i	\(0.772660\pi\)
\(29\)	66527.4	−	160611.i	0.506533	−	1.22288i	−0.439334	−	0.898324i	\(-0.644785\pi\)
							0.945867	−	0.324555i	\(-0.105215\pi\)
\(31\)	−78480.2			−0.473145			−0.236573	−	0.971614i	\(-0.576024\pi\)
							−0.236573	+	0.971614i	\(0.576024\pi\)
\(37\)	−282816.	+	117146.i	−0.917905	+	0.380209i	−0.791077	−	0.611716i	\(-0.790480\pi\)
							−0.126828	+	0.991925i	\(0.540480\pi\)
\(41\)	−345321.	−	345321.i	−0.782492	−	0.782492i	0.197759	−	0.980251i	\(-0.436634\pi\)
							−0.980251	+	0.197759i	\(0.936634\pi\)
\(43\)	−244854.	−	591131.i	−0.469644	−	1.13382i	−0.964319	−	0.264742i	\(-0.914713\pi\)
							0.494676	−	0.869078i	\(-0.335287\pi\)
\(47\)			1.10231e6i			1.54867i	0.632773	+	0.774337i	\(0.281916\pi\)
							−0.632773	+	0.774337i	\(0.718084\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.8.g.a.13.13	yes	108
32.5	even	8	inner	32.8.g.a.5.13	✓	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.8.g.a.5.13	✓	108	32.5	even	8	inner
32.8.g.a.13.13	yes	108	1.1	even	1	trivial