Embedded newform 32.9.h.a.19.12

• Label: 32.9.h.a.19.12
• 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			19.12
Character	\(\chi\)	\(=\)	32.19
Dual form			32.9.h.a.27.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.89805 + 13.9148i) q^{2} +(-80.5866 - 33.3801i) q^{3} +(-131.242 - 219.799i) q^{4} +(492.496 - 203.998i) q^{5} +(1100.95 - 857.707i) q^{6} +(729.981 + 729.981i) q^{7} +(4095.01 - 90.2160i) q^{8} +(740.648 + 740.648i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 124 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.89805	+	13.9148i	−0.493628	+	0.869673i
							\(3\)	−80.5866	−	33.3801i	−0.994897	−	0.412100i	−0.174973	−	0.984573i	\(-0.555984\pi\)
−0.819923	+	0.572473i	\(0.805984\pi\)
\(5\)	492.496	−	203.998i	0.787993	−	0.326398i	0.0478569	−	0.998854i	\(-0.484761\pi\)
							0.740137	+	0.672457i	\(0.234761\pi\)
\(7\)	729.981	+	729.981i	0.304032	+	0.304032i	0.842589	−	0.538557i	\(-0.181030\pi\)
							−0.538557	+	0.842589i	\(0.681030\pi\)
\(11\)	−6352.26	+	2631.19i	−0.433868	+	0.179714i	−0.588918	−	0.808193i	\(-0.700446\pi\)
							0.155050	+	0.987907i	\(0.450446\pi\)
\(13\)	−3948.98	−	1635.72i	−0.138265	−	0.0572712i	0.312478	−	0.949925i	\(-0.398841\pi\)
							−0.450743	+	0.892654i	\(0.648841\pi\)
\(17\)		−	62383.8i		−	0.746924i	−0.927646	−	0.373462i	\(-0.878171\pi\)
							0.927646	−	0.373462i	\(-0.121829\pi\)
\(19\)	−17461.6	+	42155.9i	−0.133989	+	0.323478i	−0.976606	−	0.215035i	\(-0.931014\pi\)
							0.842618	+	0.538512i	\(0.181014\pi\)
\(23\)	−332756.	+	332756.i	−1.18909	+	1.18909i	−0.211770	+	0.977320i	\(0.567923\pi\)
							−0.977320	+	0.211770i	\(0.932077\pi\)
\(29\)	−195160.	+	471158.i	−0.275930	+	0.666153i	−0.999715	−	0.0238719i	\(-0.992401\pi\)
							0.723785	+	0.690025i	\(0.242401\pi\)
\(31\)			1.18674e6i			1.28501i	0.766280	+	0.642507i	\(0.222106\pi\)
							−0.766280	+	0.642507i	\(0.777894\pi\)
\(37\)	170190.	−	70494.8i	0.0908084	−	0.0376141i	−0.336817	−	0.941570i	\(-0.609350\pi\)
							0.427625	+	0.903956i	\(0.359350\pi\)
\(41\)	663274.	+	663274.i	0.234724	+	0.234724i	0.814661	−	0.579937i	\(-0.196923\pi\)
							−0.579937	+	0.814661i	\(0.696923\pi\)
\(43\)	2.55373e6	−	1.05779e6i	0.746965	−	0.309403i	0.0234627	−	0.999725i	\(-0.492531\pi\)
							0.723503	+	0.690321i	\(0.242531\pi\)
\(47\)	−6.35198e6			−1.30172			−0.650861	−	0.759197i	\(-0.725592\pi\)
							−0.650861	+	0.759197i	\(0.725592\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.9.h.a.19.12	✓	124
32.27	odd	8	inner	32.9.h.a.27.12	yes	124

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.9.h.a.19.12	✓	124	1.1	even	1	trivial
32.9.h.a.27.12	yes	124	32.27	odd	8	inner