Embedded newform 32.9.h.a.19.21

• Label: 32.9.h.a.19.21
• Level: $32$
• Weight: $9$
• Character: 32.19
• Analytic conductor: $13.036$
• Analytic rank: $0$
• Dimension: $124$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	32.h (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.0361155220\)
Analytic rank:	\(0\)
Dimension:	\(124\)
Relative dimension:	\(31\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			19.21
Character	\(\chi\)	\(=\)	32.19
Dual form			32.9.h.a.27.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(8.64464 + 13.4637i) q^{2} +(114.008 + 47.2237i) q^{3} +(-106.540 + 232.777i) q^{4} +(403.645 - 167.195i) q^{5} +(349.756 + 1943.20i) q^{6} +(1058.94 + 1058.94i) q^{7} +(-4055.03 + 577.853i) q^{8} +(6128.45 + 6128.45i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 124 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	8.64464	+	13.4637i	0.540290	+	0.841479i
							\(3\)	114.008	+	47.2237i	1.40751	+	0.583009i	0.951689	−	0.307063i	\(-0.0993464\pi\)
0.455819	+	0.890072i	\(0.349346\pi\)
\(5\)	403.645	−	167.195i	0.645831	−	0.267512i	−0.0356312	−	0.999365i	\(-0.511344\pi\)
							0.681463	+	0.731853i	\(0.261344\pi\)
\(7\)	1058.94	+	1058.94i	0.441042	+	0.441042i	0.892362	−	0.451320i	\(-0.149047\pi\)
							−0.451320	+	0.892362i	\(0.649047\pi\)
\(11\)	6129.06	−	2538.74i	0.418623	−	0.173399i	−0.163421	−	0.986556i	\(-0.552253\pi\)
							0.582044	+	0.813157i	\(0.302253\pi\)
\(13\)	−5725.54	−	2371.59i	−0.200467	−	0.0830361i	0.280191	−	0.959944i	\(-0.409602\pi\)
							−0.480658	+	0.876908i	\(0.659602\pi\)
\(17\)		−	145625.i		−	1.74358i	−0.489883	−	0.871788i	\(-0.662960\pi\)
							0.489883	−	0.871788i	\(-0.337040\pi\)
\(19\)	−60717.9	+	146586.i	−0.465910	+	1.12481i	0.500022	+	0.866013i	\(0.333325\pi\)
							−0.965932	+	0.258794i	\(0.916675\pi\)
\(23\)	−144562.	+	144562.i	−0.516588	+	0.516588i	−0.916537	−	0.399949i	\(-0.869028\pi\)
							0.399949	+	0.916537i	\(0.369028\pi\)
\(29\)	274964.	−	663821.i	0.388762	−	0.938554i	−0.601441	−	0.798917i	\(-0.705407\pi\)
							0.990203	−	0.139637i	\(-0.0445934\pi\)
\(31\)			383899.i			0.415690i	0.978162	+	0.207845i	\(0.0666450\pi\)
							−0.978162	+	0.207845i	\(0.933355\pi\)
\(37\)	2.47382e6	−	1.02469e6i	1.31996	−	0.546745i	0.392185	−	0.919887i	\(-0.371719\pi\)
							0.927774	+	0.373142i	\(0.121719\pi\)
\(41\)	−2.62756e6	−	2.62756e6i	−0.929860	−	0.929860i	0.0678362	−	0.997696i	\(-0.478390\pi\)
							−0.997696	+	0.0678362i	\(0.978390\pi\)
\(43\)	4.86547e6	−	2.01534e6i	1.42315	−	0.589488i	0.467501	−	0.883993i	\(-0.345155\pi\)
							0.955650	+	0.294504i	\(0.0951545\pi\)
\(47\)	3.20913e6			0.657652			0.328826	−	0.944391i	\(-0.393347\pi\)
							0.328826	+	0.944391i	\(0.393347\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.9.h.a.19.21	✓	124
32.27	odd	8	inner	32.9.h.a.27.21	yes	124

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.9.h.a.19.21	✓	124	1.1	even	1	trivial
32.9.h.a.27.21	yes	124	32.27	odd	8	inner