Embedded newform 32.4.g.a.21.1

• Label: 32.4.g.a.21.1
• Level: $32$
• Weight: $4$
• Character: 32.21
• 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			21.1
Character	\(\chi\)	\(=\)	32.21
Dual form			32.4.g.a.29.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.73464 - 0.722312i) q^{2} +(-1.65706 + 0.686375i) q^{3} +(6.95653 + 3.95053i) q^{4} +(-4.13953 + 9.99370i) q^{5} +(5.02723 - 0.680078i) q^{6} +(24.2273 + 24.2273i) q^{7} +(-16.1701 - 15.8281i) q^{8} +(-16.8172 + 16.8172i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.73464	−	0.722312i	−0.966842	−	0.255376i
							\(3\)	−1.65706	+	0.686375i	−0.318900	+	0.132093i	−0.536391	−	0.843969i	\(-0.680213\pi\)
0.217491	+	0.976062i	\(0.430213\pi\)
\(5\)	−4.13953	+	9.99370i	−0.370251	+	0.893864i	0.623457	+	0.781858i	\(0.285728\pi\)
							−0.993708	+	0.112006i	\(0.964272\pi\)
\(7\)	24.2273	+	24.2273i	1.30815	+	1.30815i	0.922748	+	0.385403i	\(0.125938\pi\)
							0.385403	+	0.922748i	\(0.374062\pi\)
\(11\)	3.73492	+	1.54705i	0.102375	+	0.0424049i	0.433283	−	0.901258i	\(-0.357355\pi\)
							−0.330908	+	0.943663i	\(0.607355\pi\)
\(13\)	−23.2063	−	56.0248i	−0.495097	−	1.19527i	−0.952095	−	0.305802i	\(-0.901076\pi\)
							0.456999	−	0.889467i	\(-0.348924\pi\)
\(17\)			26.9981i			0.385177i	0.981280	+	0.192589i	\(0.0616883\pi\)
							−0.981280	+	0.192589i	\(0.938312\pi\)
\(19\)	22.7209	+	54.8531i	0.274344	+	0.662324i	0.999660	−	0.0260920i	\(-0.00830627\pi\)
							−0.725316	+	0.688416i	\(0.758306\pi\)
\(23\)	76.0566	−	76.0566i	0.689517	−	0.689517i	−0.272608	−	0.962125i	\(-0.587886\pi\)
							0.962125	+	0.272608i	\(0.0878861\pi\)
\(29\)	−108.152	+	44.7980i	−0.692527	+	0.286854i	−0.701053	−	0.713109i	\(-0.747286\pi\)
							0.00852531	+	0.999964i	\(0.497286\pi\)
\(31\)	176.000			1.01969			0.509846	−	0.860266i	\(-0.329702\pi\)
							0.509846	+	0.860266i	\(0.329702\pi\)
\(37\)	129.191	−	311.894i	0.574021	−	1.38581i	−0.324083	−	0.946029i	\(-0.605056\pi\)
							0.898105	−	0.439782i	\(-0.144944\pi\)
\(41\)	70.4988	−	70.4988i	0.268538	−	0.268538i	−0.559973	−	0.828511i	\(-0.689188\pi\)
							0.828511	+	0.559973i	\(0.189188\pi\)
\(43\)	−103.155	−	42.7281i	−0.365836	−	0.151534i	0.192190	−	0.981358i	\(-0.438441\pi\)
							−0.558026	+	0.829824i	\(0.688441\pi\)
\(47\)		−	249.437i		−	0.774131i	−0.922052	−	0.387066i	\(-0.873489\pi\)
							0.922052	−	0.387066i	\(-0.126511\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.4.g.a.21.1	✓	44
4.3	odd	2		128.4.g.a.49.7		44
8.3	odd	2		256.4.g.a.97.5		44
8.5	even	2		256.4.g.b.97.7		44
32.3	odd	8		128.4.g.a.81.7		44
32.13	even	8		256.4.g.b.161.7		44
32.19	odd	8		256.4.g.a.161.5		44
32.29	even	8	inner	32.4.g.a.29.1	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.4.g.a.21.1	✓	44	1.1	even	1	trivial
32.4.g.a.29.1	yes	44	32.29	even	8	inner
128.4.g.a.49.7		44	4.3	odd	2
128.4.g.a.81.7		44	32.3	odd	8
256.4.g.a.97.5		44	8.3	odd	2
256.4.g.a.161.5		44	32.19	odd	8
256.4.g.b.97.7		44	8.5	even	2
256.4.g.b.161.7		44	32.13	even	8