Embedded newform 32.4.g.a.5.3

• Label: 32.4.g.a.5.3
• Level: $32$
• Weight: $4$
• Character: 32.5
• Analytic conductor: $1.888$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	32.g (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.88806112018\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(11\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			5.3
Character	\(\chi\)	\(=\)	32.5
Dual form			32.4.g.a.13.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.10117 - 1.89344i) q^{2} +(1.94138 + 4.68690i) q^{3} +(0.829801 + 7.95685i) q^{4} +(4.93338 + 2.04347i) q^{5} +(4.79518 - 13.5238i) q^{6} +(14.0755 + 14.0755i) q^{7} +(13.3222 - 18.2898i) q^{8} +(0.893826 - 0.893826i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{1}{8}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.10117	−	1.89344i	−0.742875	−	0.669431i
							\(3\)	1.94138	+	4.68690i	0.373618	+	0.901994i	0.993131	+	0.117007i	\(0.0373301\pi\)
−0.619513	+	0.784986i	\(0.712670\pi\)
\(5\)	4.93338	+	2.04347i	0.441255	+	0.182774i	0.592239	−	0.805762i	\(-0.298244\pi\)
							−0.150984	+	0.988536i	\(0.548244\pi\)
\(7\)	14.0755	+	14.0755i	0.760005	+	0.760005i	0.976323	−	0.216318i	\(-0.0694049\pi\)
							−0.216318	+	0.976323i	\(0.569405\pi\)
\(11\)	−3.78733	+	9.14343i	−0.103811	+	0.250623i	−0.967247	−	0.253836i	\(-0.918308\pi\)
							0.863436	+	0.504458i	\(0.168308\pi\)
\(13\)	−64.7407	+	26.8165i	−1.38122	+	0.572120i	−0.944807	−	0.327627i	\(-0.893751\pi\)
							−0.436412	+	0.899747i	\(0.643751\pi\)
\(17\)		−	79.3923i		−	1.13267i	−0.824174	−	0.566337i	\(-0.808360\pi\)
							0.824174	−	0.566337i	\(-0.191640\pi\)
\(19\)	94.7756	−	39.2573i	1.14437	−	0.474013i	0.271727	−	0.962374i	\(-0.412405\pi\)
							0.872642	+	0.488361i	\(0.162405\pi\)
\(23\)	71.6801	−	71.6801i	0.649841	−	0.649841i	−0.303114	−	0.952954i	\(-0.598026\pi\)
							0.952954	+	0.303114i	\(0.0980261\pi\)
\(29\)	−53.0409	−	128.052i	−0.339636	−	0.819955i	−0.997751	−	0.0670366i	\(-0.978646\pi\)
							0.658114	−	0.752918i	\(-0.271354\pi\)
\(31\)	−267.650			−1.55069			−0.775346	−	0.631537i	\(-0.782424\pi\)
							−0.775346	+	0.631537i	\(0.782424\pi\)
\(37\)	205.678	+	85.1946i	0.913872	+	0.378538i	0.789537	−	0.613702i	\(-0.210321\pi\)
							0.124334	+	0.992240i	\(0.460321\pi\)
\(41\)	210.468	−	210.468i	0.801697	−	0.801697i	−0.181664	−	0.983361i	\(-0.558148\pi\)
							0.983361	+	0.181664i	\(0.0581482\pi\)
\(43\)	−56.9749	+	137.550i	−0.202060	+	0.487817i	−0.992132	−	0.125198i	\(-0.960043\pi\)
							0.790072	+	0.613015i	\(0.210043\pi\)
\(47\)			173.739i			0.539201i	0.962972	+	0.269600i	\(0.0868916\pi\)
							−0.962972	+	0.269600i	\(0.913108\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.4.g.a.5.3	✓	44
4.3	odd	2		128.4.g.a.113.3		44
8.3	odd	2		256.4.g.a.225.9		44
8.5	even	2		256.4.g.b.225.3		44
32.3	odd	8		256.4.g.a.33.9		44
32.13	even	8	inner	32.4.g.a.13.3	yes	44
32.19	odd	8		128.4.g.a.17.3		44
32.29	even	8		256.4.g.b.33.3		44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.4.g.a.5.3	✓	44	1.1	even	1	trivial
32.4.g.a.13.3	yes	44	32.13	even	8	inner
128.4.g.a.17.3		44	32.19	odd	8
128.4.g.a.113.3		44	4.3	odd	2
256.4.g.a.33.9		44	32.3	odd	8
256.4.g.a.225.9		44	8.3	odd	2
256.4.g.b.33.3		44	32.29	even	8
256.4.g.b.225.3		44	8.5	even	2