Embedded newform 32.9.h.a.19.30

• Label: 32.9.h.a.19.30
• 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			19.30
Character	\(\chi\)	\(=\)	32.19
Dual form			32.9.h.a.27.30

$q$-expansion

\(f(q)\)	\(=\)	\(q+(15.9987 - 0.207777i) q^{2} +(114.955 + 47.6159i) q^{3} +(255.914 - 6.64829i) q^{4} +(-373.757 + 154.815i) q^{5} +(1849.02 + 737.905i) q^{6} +(929.082 + 929.082i) q^{7} +(4092.89 - 159.537i) q^{8} +(6308.04 + 6308.04i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 124 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	15.9987	−	0.207777i	0.999916	−	0.0129860i
							\(3\)	114.955	+	47.6159i	1.41920	+	0.587851i	0.954659	−	0.297702i	\(-0.0962203\pi\)
0.464538	+	0.885553i	\(0.346220\pi\)
\(5\)	−373.757	+	154.815i	−0.598011	+	0.247704i	−0.661093	−	0.750304i	\(-0.729907\pi\)
							0.0630817	+	0.998008i	\(0.479907\pi\)
\(7\)	929.082	+	929.082i	0.386956	+	0.386956i	0.873600	−	0.486644i	\(-0.161779\pi\)
							−0.486644	+	0.873600i	\(0.661779\pi\)
\(11\)	6384.99	−	2644.75i	0.436104	−	0.180640i	−0.153820	−	0.988099i	\(-0.549158\pi\)
							0.589924	+	0.807459i	\(0.299158\pi\)
\(13\)	−47284.8	−	19586.0i	−1.65557	−	0.685761i	−0.657847	−	0.753152i	\(-0.728533\pi\)
							−0.997727	+	0.0673908i	\(0.978533\pi\)
\(17\)			102962.i			1.23277i	0.787446	+	0.616383i	\(0.211403\pi\)
							−0.787446	+	0.616383i	\(0.788597\pi\)
\(19\)	73360.0	−	177107.i	0.562918	−	1.35900i	−0.344505	−	0.938785i	\(-0.611953\pi\)
							0.907422	−	0.420219i	\(-0.138047\pi\)
\(23\)	183193.	−	183193.i	0.654633	−	0.654633i	−0.299472	−	0.954105i	\(-0.596811\pi\)
							0.954105	+	0.299472i	\(0.0968106\pi\)
\(29\)	94615.5	−	228422.i	0.133774	−	0.322958i	−0.842771	−	0.538272i	\(-0.819077\pi\)
							0.976545	+	0.215314i	\(0.0690774\pi\)
\(31\)		−	39506.9i		−	0.0427786i	−0.999771	−	0.0213893i	\(-0.993191\pi\)
							0.999771	−	0.0213893i	\(-0.00680894\pi\)
\(37\)	−1.62788e6	+	674289.i	−0.868591	+	0.359782i	−0.772061	−	0.635548i	\(-0.780774\pi\)
							−0.0965294	+	0.995330i	\(0.530774\pi\)
\(41\)	−2.04944e6	−	2.04944e6i	−0.725271	−	0.725271i	0.244403	−	0.969674i	\(-0.421408\pi\)
							−0.969674	+	0.244403i	\(0.921408\pi\)
\(43\)	1.24193e6	−	514423.i	0.363264	−	0.150469i	−0.193583	−	0.981084i	\(-0.562011\pi\)
							0.556847	+	0.830615i	\(0.312011\pi\)
\(47\)	−4.29832e6			−0.880862			−0.440431	−	0.897787i	\(-0.645174\pi\)
							−0.440431	+	0.897787i	\(0.645174\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.9.h.a.19.30	✓	124
32.27	odd	8	inner	32.9.h.a.27.30	yes	124

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.9.h.a.19.30	✓	124	1.1	even	1	trivial
32.9.h.a.27.30	yes	124	32.27	odd	8	inner