Embedded newform 32.8.g.a.13.17

• Label: 32.8.g.a.13.17
• 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			13.17
Character	\(\chi\)	\(=\)	32.13
Dual form			32.8.g.a.5.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.26269 + 11.0851i) q^{2} +(-17.9118 + 43.2428i) q^{3} +(-117.760 + 50.1645i) q^{4} +(-367.733 + 152.320i) q^{5} +(-519.882 - 100.709i) q^{6} +(951.213 - 951.213i) q^{7} +(-822.535 - 1191.88i) q^{8} +(-2.66939 - 2.66939i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.26269	+	11.0851i	0.199995	+	0.979797i
							\(3\)	−17.9118	+	43.2428i	−0.383014	+	0.924677i	0.608366	+	0.793656i	\(0.291825\pi\)
−0.991380	+	0.131020i	\(0.958175\pi\)
\(5\)	−367.733	+	152.320i	−1.31564	+	0.544957i	−0.926526	−	0.376232i	\(-0.877220\pi\)
							−0.389117	+	0.921188i	\(0.627220\pi\)
\(7\)	951.213	−	951.213i	1.04818	−	1.04818i	0.0493984	−	0.998779i	\(-0.484270\pi\)
							0.998779	−	0.0493984i	\(-0.0157304\pi\)
\(11\)	2609.97	+	6301.02i	0.591236	+	1.42737i	0.882309	+	0.470670i	\(0.155988\pi\)
							−0.291073	+	0.956701i	\(0.594012\pi\)
\(13\)	−7367.94	−	3051.90i	−0.930132	−	0.385273i	−0.134403	−	0.990927i	\(-0.542912\pi\)
							−0.795729	+	0.605653i	\(0.792912\pi\)
\(17\)		−	7298.45i		−	0.360296i	−0.983640	−	0.180148i	\(-0.942342\pi\)
							0.983640	−	0.180148i	\(-0.0576577\pi\)
\(19\)	−21378.7	−	8855.34i	−0.715061	−	0.296188i	−0.00466404	−	0.999989i	\(-0.501485\pi\)
							−0.710397	+	0.703801i	\(0.751485\pi\)
\(23\)	−59201.3	−	59201.3i	−1.01457	−	1.01457i	−0.999892	−	0.0146823i	\(-0.995326\pi\)
							−0.0146823	−	0.999892i	\(-0.504674\pi\)
\(29\)	−5135.39	+	12397.9i	−0.0391003	+	0.0943965i	−0.942223	−	0.334985i	\(-0.891269\pi\)
							0.903123	+	0.429382i	\(0.141269\pi\)
\(31\)	−150772.			−0.908984			−0.454492	−	0.890751i	\(-0.650179\pi\)
							−0.454492	+	0.890751i	\(0.650179\pi\)
\(37\)	−84887.3	+	35161.5i	−0.275510	+	0.114120i	−0.516160	−	0.856492i	\(-0.672639\pi\)
							0.240651	+	0.970612i	\(0.422639\pi\)
\(41\)	512409.	+	512409.i	1.16111	+	1.16111i	0.984233	+	0.176878i	\(0.0565997\pi\)
							0.176878	+	0.984233i	\(0.443400\pi\)
\(43\)	−56397.6	−	136156.i	−0.108174	−	0.261154i	0.860519	−	0.509419i	\(-0.170140\pi\)
							−0.968692	+	0.248265i	\(0.920140\pi\)
\(47\)			1.26610e6i			1.77879i	0.457140	+	0.889395i	\(0.348874\pi\)
							−0.457140	+	0.889395i	\(0.651126\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.8.g.a.13.17	yes	108
32.5	even	8	inner	32.8.g.a.5.17	✓	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.8.g.a.5.17	✓	108	32.5	even	8	inner
32.8.g.a.13.17	yes	108	1.1	even	1	trivial