Embedded newform 32.4.g.a.5.4

• Label: 32.4.g.a.5.4
• 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			5.4
Character	\(\chi\)	\(=\)	32.5
Dual form			32.4.g.a.13.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.75970 + 2.21438i) q^{2} +(3.54796 + 8.56554i) q^{3} +(-1.80692 - 7.79327i) q^{4} +(-7.55322 - 3.12865i) q^{5} +(-25.2107 - 7.21625i) q^{6} +(7.16166 + 7.16166i) q^{7} +(20.4369 + 9.71261i) q^{8} +(-41.6886 + 41.6886i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.75970	+	2.21438i	−0.622148	+	0.782900i
							\(3\)	3.54796	+	8.56554i	0.682806	+	1.64844i	0.758794	+	0.651331i	\(0.225789\pi\)
−0.0759878	+	0.997109i	\(0.524211\pi\)
\(5\)	−7.55322	−	3.12865i	−0.675581	−	0.279835i	0.0183975	−	0.999831i	\(-0.494144\pi\)
							−0.693978	+	0.719996i	\(0.744144\pi\)
\(7\)	7.16166	+	7.16166i	0.386693	+	0.386693i	0.873506	−	0.486813i	\(-0.161841\pi\)
							−0.486813	+	0.873506i	\(0.661841\pi\)
\(11\)	0.758120	−	1.83026i	0.0207802	−	0.0501678i	−0.913149	−	0.407626i	\(-0.866357\pi\)
							0.933929	+	0.357458i	\(0.116357\pi\)
\(13\)	71.0832	−	29.4436i	1.51653	−	0.628168i	0.539639	−	0.841896i	\(-0.318561\pi\)
							0.976893	+	0.213728i	\(0.0685606\pi\)
\(17\)			98.5470i			1.40595i	0.711214	+	0.702975i	\(0.248146\pi\)
							−0.711214	+	0.702975i	\(0.751854\pi\)
\(19\)	89.5748	−	37.1031i	1.08157	−	0.448002i	0.230512	−	0.973070i	\(-0.425960\pi\)
							0.851060	+	0.525068i	\(0.175960\pi\)
\(23\)	24.9355	−	24.9355i	0.226062	−	0.226062i	−0.584984	−	0.811045i	\(-0.698899\pi\)
							0.811045	+	0.584984i	\(0.198899\pi\)
\(29\)	−57.8528	−	139.669i	−0.370448	−	0.894341i	−0.993674	−	0.112299i	\(-0.964178\pi\)
							0.623226	−	0.782042i	\(-0.285822\pi\)
\(31\)	58.0545			0.336351			0.168176	−	0.985757i	\(-0.446212\pi\)
							0.168176	+	0.985757i	\(0.446212\pi\)
\(37\)	−202.968	−	84.0720i	−0.901829	−	0.373550i	−0.116906	−	0.993143i	\(-0.537298\pi\)
							−0.784923	+	0.619593i	\(0.787298\pi\)
\(41\)	−45.3618	+	45.3618i	−0.172788	+	0.172788i	−0.788203	−	0.615415i	\(-0.788988\pi\)
							0.615415	+	0.788203i	\(0.288988\pi\)
\(43\)	89.7175	−	216.597i	0.318181	−	0.768157i	−0.681170	−	0.732126i	\(-0.738528\pi\)
							0.999351	−	0.0360314i	\(-0.0114716\pi\)
\(47\)			4.38416i			0.0136063i	0.999977	+	0.00680315i	\(0.00216553\pi\)
							−0.999977	+	0.00680315i	\(0.997834\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.4.g.a.5.4	✓	44
4.3	odd	2		128.4.g.a.113.1		44
8.3	odd	2		256.4.g.a.225.11		44
8.5	even	2		256.4.g.b.225.1		44
32.3	odd	8		256.4.g.a.33.11		44
32.13	even	8	inner	32.4.g.a.13.4	yes	44
32.19	odd	8		128.4.g.a.17.1		44
32.29	even	8		256.4.g.b.33.1		44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.4.g.a.5.4	✓	44	1.1	even	1	trivial
32.4.g.a.13.4	yes	44	32.13	even	8	inner
128.4.g.a.17.1		44	32.19	odd	8
128.4.g.a.113.1		44	4.3	odd	2
256.4.g.a.33.11		44	32.3	odd	8
256.4.g.a.225.11		44	8.3	odd	2
256.4.g.b.33.1		44	32.29	even	8
256.4.g.b.225.1		44	8.5	even	2