Embedded newform 32.9.h.a.19.1

• Label: 32.9.h.a.19.1
• 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			19.1
Character	\(\chi\)	\(=\)	32.19
Dual form			32.9.h.a.27.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-15.9992 + 0.154928i) q^{2} +(123.520 + 51.1638i) q^{3} +(255.952 - 4.95748i) q^{4} +(-777.947 + 322.236i) q^{5} +(-1984.16 - 799.446i) q^{6} +(-2246.92 - 2246.92i) q^{7} +(-4094.27 + 118.970i) q^{8} +(8000.23 + 8000.23i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 124 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−15.9992	+	0.154928i	−0.999953	+	0.00968303i
							\(3\)	123.520	+	51.1638i	1.52494	+	0.631652i	0.978575	−	0.205892i	\(-0.0660096\pi\)
0.546369	+	0.837545i	\(0.316010\pi\)
\(5\)	−777.947	+	322.236i	−1.24472	+	0.515578i	−0.905185	−	0.425017i	\(-0.860268\pi\)
							−0.339530	+	0.940595i	\(0.610268\pi\)
\(7\)	−2246.92	−	2246.92i	−0.935825	−	0.935825i	0.0622367	−	0.998061i	\(-0.480177\pi\)
							−0.998061	+	0.0622367i	\(0.980177\pi\)
\(11\)	−2816.62	+	1166.68i	−0.192379	+	0.0796859i	−0.476793	−	0.879016i	\(-0.658201\pi\)
							0.284414	+	0.958701i	\(0.408201\pi\)
\(13\)	−22825.3	−	9454.57i	−0.799179	−	0.331031i	−0.0545514	−	0.998511i	\(-0.517373\pi\)
							−0.744628	+	0.667480i	\(0.767373\pi\)
\(17\)		−	116238.i		−	1.39172i	−0.718177	−	0.695861i	\(-0.755023\pi\)
							0.718177	−	0.695861i	\(-0.244977\pi\)
\(19\)	8130.86	−	19629.6i	0.0623910	−	0.150625i	−0.889609	−	0.456722i	\(-0.849023\pi\)
							0.952000	+	0.306097i	\(0.0990233\pi\)
\(23\)	−375527.	+	375527.i	−1.34193	+	1.34193i	−0.447789	+	0.894139i	\(0.647789\pi\)
							−0.894139	+	0.447789i	\(0.852211\pi\)
\(29\)	−108403.	+	261709.i	−0.153268	+	0.370021i	−0.981799	−	0.189921i	\(-0.939177\pi\)
							0.828532	+	0.559942i	\(0.189177\pi\)
\(31\)		−	1.57416e6i		−	1.70452i	−0.523118	−	0.852260i	\(-0.675231\pi\)
							0.523118	−	0.852260i	\(-0.324769\pi\)
\(37\)	236950.	−	98148.1i	0.126430	−	0.0523691i	−0.318571	−	0.947899i	\(-0.603203\pi\)
							0.445002	+	0.895530i	\(0.353203\pi\)
\(41\)	−657845.	−	657845.i	−0.232803	−	0.232803i	0.581059	−	0.813862i	\(-0.302639\pi\)
							−0.813862	+	0.581059i	\(0.802639\pi\)
\(43\)	−5.17600e6	+	2.14397e6i	−1.51398	+	0.627112i	−0.976375	−	0.216083i	\(-0.930672\pi\)
							−0.537607	+	0.843195i	\(0.680672\pi\)
\(47\)	−1.64708e6			−0.337538			−0.168769	−	0.985656i	\(-0.553979\pi\)
							−0.168769	+	0.985656i	\(0.553979\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.9.h.a.19.1	✓	124
32.27	odd	8	inner	32.9.h.a.27.1	yes	124

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.9.h.a.19.1	✓	124	1.1	even	1	trivial
32.9.h.a.27.1	yes	124	32.27	odd	8	inner