Embedded newform 32.9.h.a.11.15

• Label: 32.9.h.a.11.15
• 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			11.15
Character	\(\chi\)	\(=\)	32.11
Dual form			32.9.h.a.3.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.79560 - 15.7539i) q^{2} +(-8.40700 - 20.2963i) q^{3} +(-240.369 + 88.0830i) q^{4} +(67.8482 - 163.800i) q^{5} +(-296.243 + 189.183i) q^{6} +(-2083.94 + 2083.94i) q^{7} +(2059.62 + 3540.50i) q^{8} +(4298.07 - 4298.07i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 124 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.79560	−	15.7539i	−0.174725	−	0.984617i
							\(3\)	−8.40700	−	20.2963i	−0.103790	−	0.250572i	0.863450	−	0.504434i	\(-0.168299\pi\)
−0.967240	+	0.253863i	\(0.918299\pi\)
\(5\)	67.8482	−	163.800i	0.108557	−	0.262080i	−0.860261	−	0.509855i	\(-0.829699\pi\)
							0.968818	+	0.247774i	\(0.0796992\pi\)
\(7\)	−2083.94	+	2083.94i	−0.867948	+	0.867948i	−0.992245	−	0.124297i	\(-0.960332\pi\)
							0.124297	+	0.992245i	\(0.460332\pi\)
\(11\)	−6449.96	+	15571.6i	−0.440541	+	1.06356i	0.535218	+	0.844714i	\(0.320229\pi\)
							−0.975759	+	0.218846i	\(0.929771\pi\)
\(13\)	7157.88	+	17280.6i	0.250617	+	0.605043i	0.998254	−	0.0590643i	\(-0.0188117\pi\)
							−0.747637	+	0.664108i	\(0.768812\pi\)
\(17\)		−	31737.4i		−	0.379993i	−0.981785	−	0.189997i	\(-0.939152\pi\)
							0.981785	−	0.189997i	\(-0.0608477\pi\)
\(19\)	146714.	−	60770.8i	1.12579	−	0.466316i	0.259441	−	0.965759i	\(-0.416462\pi\)
							0.866347	+	0.499443i	\(0.166462\pi\)
\(23\)	200270.	+	200270.i	0.715656	+	0.715656i	0.967712	−	0.252057i	\(-0.0811070\pi\)
							−0.252057	+	0.967712i	\(0.581107\pi\)
\(29\)	−801780.	+	332108.i	−1.13361	+	0.469556i	−0.869006	−	0.494801i	\(-0.835241\pi\)
							−0.264603	+	0.964357i	\(0.585241\pi\)
\(31\)			1.21688e6i			1.31766i	0.752293	+	0.658828i	\(0.228948\pi\)
							−0.752293	+	0.658828i	\(0.771052\pi\)
\(37\)	350991.	−	847368.i	0.187279	−	0.452132i	−0.802155	−	0.597116i	\(-0.796313\pi\)
							0.989434	+	0.144984i	\(0.0463132\pi\)
\(41\)	−2.86327e6	+	2.86327e6i	−1.01327	+	1.01327i	−0.0133623	+	0.999911i	\(0.504253\pi\)
							−0.999911	+	0.0133623i	\(0.995747\pi\)
\(43\)	−1.75976e6	+	4.24843e6i	−0.514729	+	1.24267i	0.426374	+	0.904547i	\(0.359791\pi\)
							−0.941103	+	0.338119i	\(0.890209\pi\)
\(47\)	−4.39311e6			−0.900287			−0.450143	−	0.892956i	\(-0.648627\pi\)
							−0.450143	+	0.892956i	\(0.648627\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.9.h.a.11.15	yes	124
32.3	odd	8	inner	32.9.h.a.3.15	✓	124

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.9.h.a.3.15	✓	124	32.3	odd	8	inner
32.9.h.a.11.15	yes	124	1.1	even	1	trivial