Embedded newform 32.8.g.a.21.26

• Label: 32.8.g.a.21.26
• Level: $32$
• Weight: $8$
• Character: 32.21
• 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			21.26
Character	\(\chi\)	\(=\)	32.21
Dual form			32.8.g.a.29.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(11.2490 - 1.20818i) q^{2} +(62.1168 - 25.7296i) q^{3} +(125.081 - 27.1817i) q^{4} +(-146.228 + 353.025i) q^{5} +(667.667 - 364.482i) q^{6} +(953.509 + 953.509i) q^{7} +(1374.19 - 456.888i) q^{8} +(1650.04 - 1650.04i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	11.2490	−	1.20818i	0.994282	−	0.106789i
							\(3\)	62.1168	−	25.7296i	1.32827	−	0.550186i	0.398105	−	0.917340i	\(-0.369668\pi\)
0.930160	+	0.367154i	\(0.119668\pi\)
\(5\)	−146.228	+	353.025i	−0.523161	+	1.26302i	0.412769	+	0.910836i	\(0.364562\pi\)
							−0.935930	+	0.352186i	\(0.885438\pi\)
\(7\)	953.509	+	953.509i	1.05071	+	1.05071i	0.998644	+	0.0520634i	\(0.0165798\pi\)
							0.0520634	+	0.998644i	\(0.483420\pi\)
\(11\)	−5837.69	−	2418.05i	−1.32241	−	0.547761i	−0.393931	−	0.919140i	\(-0.628885\pi\)
							−0.928481	+	0.371379i	\(0.878885\pi\)
\(13\)	−5161.18	−	12460.2i	−0.651549	−	1.57298i	−0.810530	−	0.585698i	\(-0.800821\pi\)
							0.158980	−	0.987282i	\(-0.449179\pi\)
\(17\)		−	10959.9i		−	0.541046i	−0.962714	−	0.270523i	\(-0.912803\pi\)
							0.962714	−	0.270523i	\(-0.0871966\pi\)
\(19\)	−4596.07	−	11095.9i	−0.153727	−	0.371129i	0.828189	−	0.560449i	\(-0.189371\pi\)
							−0.981915	+	0.189321i	\(0.939371\pi\)
\(23\)	−30332.2	+	30332.2i	−0.519823	+	0.519823i	−0.917518	−	0.397694i	\(-0.869810\pi\)
							0.397694	+	0.917518i	\(0.369810\pi\)
\(29\)	−118475.	+	49074.0i	−0.902057	+	0.373644i	−0.785011	−	0.619482i	\(-0.787343\pi\)
							−0.117047	+	0.993126i	\(0.537343\pi\)
\(31\)	240122.			1.44766			0.723830	−	0.689979i	\(-0.242380\pi\)
							0.723830	+	0.689979i	\(0.242380\pi\)
\(37\)	−1432.88	+	3459.28i	−0.00465055	+	0.0112274i	−0.926188	−	0.377063i	\(-0.876934\pi\)
							0.921537	+	0.388290i	\(0.126934\pi\)
\(41\)	−53713.5	+	53713.5i	−0.121714	+	0.121714i	−0.765340	−	0.643626i	\(-0.777429\pi\)
							0.643626	+	0.765340i	\(0.277429\pi\)
\(43\)	−170184.	−	70492.6i	−0.326422	−	0.135209i	0.213454	−	0.976953i	\(-0.431529\pi\)
							−0.539876	+	0.841745i	\(0.681529\pi\)
\(47\)		−	116511.i		−	0.163691i	−0.996645	−	0.0818456i	\(-0.973919\pi\)
							0.996645	−	0.0818456i	\(-0.0260814\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.8.g.a.21.26	✓	108
32.29	even	8	inner	32.8.g.a.29.26	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.8.g.a.21.26	✓	108	1.1	even	1	trivial
32.8.g.a.29.26	yes	108	32.29	even	8	inner