Embedded newform 32.9.h.a.11.27

• Label: 32.9.h.a.11.27
• 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			11.27
Character	\(\chi\)	\(=\)	32.11
Dual form			32.9.h.a.3.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(14.7296 + 6.24809i) q^{2} +(14.2255 + 34.3434i) q^{3} +(177.923 + 184.064i) q^{4} +(-33.8815 + 81.7973i) q^{5} +(-5.04461 + 594.748i) q^{6} +(-2382.85 + 2382.85i) q^{7} +(1470.68 + 3822.87i) q^{8} +(3662.22 - 3662.22i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 124 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	14.7296	+	6.24809i	0.920601	+	0.390506i
							\(3\)	14.2255	+	34.3434i	0.175624	+	0.423993i	0.987040	−	0.160476i	\(-0.0513029\pi\)
−0.811416	+	0.584469i	\(0.801303\pi\)
\(5\)	−33.8815	+	81.7973i	−0.0542105	+	0.130876i	−0.948664	−	0.316284i	\(-0.897565\pi\)
							0.894454	+	0.447160i	\(0.147565\pi\)
\(7\)	−2382.85	+	2382.85i	−0.992442	+	0.992442i	−0.999972	−	0.00752934i	\(-0.997603\pi\)
							0.00752934	+	0.999972i	\(0.497603\pi\)
\(11\)	−4481.27	+	10818.7i	−0.306077	+	0.738934i	0.693748	+	0.720218i	\(0.255958\pi\)
							−0.999825	+	0.0187165i	\(0.994042\pi\)
\(13\)	−1848.21	−	4461.98i	−0.0647111	−	0.156226i	0.888216	−	0.459426i	\(-0.151945\pi\)
							−0.952927	+	0.303200i	\(0.901945\pi\)
\(17\)			52612.9i			0.629936i	0.949102	+	0.314968i	\(0.101994\pi\)
							−0.949102	+	0.314968i	\(0.898006\pi\)
\(19\)	8546.85	−	3540.22i	0.0655831	−	0.0271654i	−0.349651	−	0.936880i	\(-0.613700\pi\)
							0.415234	+	0.909715i	\(0.363700\pi\)
\(23\)	−168811.	−	168811.i	−0.603240	−	0.603240i	0.337931	−	0.941171i	\(-0.390273\pi\)
							−0.941171	+	0.337931i	\(0.890273\pi\)
\(29\)	606672.	−	251292.i	0.857752	−	0.355293i	0.0899241	−	0.995949i	\(-0.471338\pi\)
							0.767828	+	0.640656i	\(0.221338\pi\)
\(31\)			414985.i			0.449351i	0.974434	+	0.224676i	\(0.0721322\pi\)
							−0.974434	+	0.224676i	\(0.927868\pi\)
\(37\)	532604.	−	1.28582e6i	0.284183	−	0.686078i	−0.715742	−	0.698365i	\(-0.753911\pi\)
							0.999925	+	0.0122872i	\(0.00391122\pi\)
\(41\)	3.86454e6	−	3.86454e6i	1.36761	−	1.36761i	0.503777	−	0.863834i	\(-0.331943\pi\)
							0.863834	−	0.503777i	\(-0.168057\pi\)
\(43\)	1.30483e6	−	3.15015e6i	0.381664	−	0.921419i	−0.609980	−	0.792417i	\(-0.708823\pi\)
							0.991644	−	0.129002i	\(-0.0411774\pi\)
\(47\)	8.55116e6			1.75240			0.876201	−	0.481947i	\(-0.160070\pi\)
							0.876201	+	0.481947i	\(0.160070\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.9.h.a.11.27	yes	124
32.3	odd	8	inner	32.9.h.a.3.27	✓	124

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.9.h.a.3.27	✓	124	32.3	odd	8	inner
32.9.h.a.11.27	yes	124	1.1	even	1	trivial