Embedded newform 32.9.h.a.11.31

• Label: 32.9.h.a.11.31
• Level: $32$
• Weight: $9$
• Character: 32.11
• 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			11.31
Character	\(\chi\)	\(=\)	32.11
Dual form			32.9.h.a.3.31

$q$-expansion

\(f(q)\)	\(=\)	\(q+(15.9617 + 1.10593i) q^{2} +(-54.6648 - 131.973i) q^{3} +(253.554 + 35.3052i) q^{4} +(440.360 - 1063.12i) q^{5} +(-726.593 - 2166.97i) q^{6} +(-1839.02 + 1839.02i) q^{7} +(4008.11 + 843.945i) q^{8} +(-9789.19 + 9789.19i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 124 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	15.9617	+	1.10593i	0.997608	+	0.0691207i
							\(3\)	−54.6648	−	131.973i	−0.674874	−	1.62929i	−0.773219	−	0.634139i	\(-0.781355\pi\)
0.0983449	−	0.995152i	\(-0.468645\pi\)
\(5\)	440.360	−	1063.12i	0.704576	−	1.70100i	−0.00855239	−	0.999963i	\(-0.502722\pi\)
							0.713128	−	0.701033i	\(-0.247278\pi\)
\(7\)	−1839.02	+	1839.02i	−0.765940	+	0.765940i	−0.977389	−	0.211449i	\(-0.932182\pi\)
							0.211449	+	0.977389i	\(0.432182\pi\)
\(11\)	7.45570	−	17.9996i	0.000509234	−	0.00122940i	−0.923625	−	0.383298i	\(-0.874788\pi\)
							0.924134	+	0.382069i	\(0.124788\pi\)
\(13\)	243.246	+	587.247i	0.00851671	+	0.0205612i	0.928081	−	0.372379i	\(-0.121458\pi\)
							−0.919564	+	0.392940i	\(0.871458\pi\)
\(17\)		−	112601.i		−	1.34818i	−0.738649	−	0.674090i	\(-0.764536\pi\)
							0.738649	−	0.674090i	\(-0.235464\pi\)
\(19\)	9248.59	−	3830.89i	0.0709678	−	0.0293958i	−0.346917	−	0.937896i	\(-0.612772\pi\)
							0.417885	+	0.908500i	\(0.362772\pi\)
\(23\)	231355.	+	231355.i	0.826736	+	0.826736i	0.987064	−	0.160328i	\(-0.0512551\pi\)
							−0.160328	+	0.987064i	\(0.551255\pi\)
\(29\)	−373835.	+	154847.i	−0.528552	+	0.218933i	−0.630969	−	0.775808i	\(-0.717342\pi\)
							0.102417	+	0.994742i	\(0.467342\pi\)
\(31\)			124710.i			0.135038i	0.997718	+	0.0675189i	\(0.0215083\pi\)
							−0.997718	+	0.0675189i	\(0.978492\pi\)
\(37\)	1.33153e6	−	3.21461e6i	0.710469	−	1.71522i	0.0116436	−	0.999932i	\(-0.496294\pi\)
							0.698826	−	0.715292i	\(-0.253706\pi\)
\(41\)	1.15513e6	−	1.15513e6i	0.408787	−	0.408787i	−0.472529	−	0.881315i	\(-0.656659\pi\)
							0.881315	+	0.472529i	\(0.156659\pi\)
\(43\)	−114613.	+	276701.i	−0.0335244	+	0.0809350i	−0.939755	−	0.341849i	\(-0.888947\pi\)
							0.906231	+	0.422784i	\(0.138947\pi\)
\(47\)	3.35944e6			0.688455			0.344227	−	0.938886i	\(-0.388141\pi\)
							0.344227	+	0.938886i	\(0.388141\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.9.h.a.11.31	yes	124
32.3	odd	8	inner	32.9.h.a.3.31	✓	124

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.9.h.a.3.31	✓	124	32.3	odd	8	inner
32.9.h.a.11.31	yes	124	1.1	even	1	trivial