Embedded newform 32.6.g.a.5.11

• Label: 32.6.g.a.5.11
• Level: $32$
• Weight: $6$
• Character: 32.5
• Analytic conductor: $5.132$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	32.g (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.13228223402\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(19\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			5.11
Character	\(\chi\)	\(=\)	32.5
Dual form			32.6.g.a.13.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.01888 + 5.56434i) q^{2} +(4.68811 + 11.3181i) q^{3} +(-29.9238 + 11.3388i) q^{4} +(-98.5667 - 40.8277i) q^{5} +(-58.2012 + 37.6180i) q^{6} +(104.908 + 104.908i) q^{7} +(-93.5814 - 154.953i) q^{8} +(65.7060 - 65.7060i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.01888	+	5.56434i	0.180114	+	0.983646i
							\(3\)	4.68811	+	11.3181i	0.300742	+	0.726056i	0.999938	+	0.0111261i	\(0.00354161\pi\)
−0.699196	+	0.714930i	\(0.746458\pi\)
\(5\)	−98.5667	−	40.8277i	−1.76322	−	0.730348i	−0.996040	−	0.0889105i	\(-0.971661\pi\)
							−0.767176	−	0.641437i	\(-0.778339\pi\)
\(7\)	104.908	+	104.908i	0.809215	+	0.809215i	0.984515	−	0.175300i	\(-0.0560895\pi\)
							−0.175300	+	0.984515i	\(0.556090\pi\)
\(11\)	−205.402	+	495.884i	−0.511827	+	1.23566i	0.430993	+	0.902355i	\(0.358164\pi\)
							−0.942819	+	0.333304i	\(0.891836\pi\)
\(13\)	−601.914	+	249.321i	−0.987816	+	0.409167i	−0.817315	−	0.576191i	\(-0.804539\pi\)
							−0.170501	+	0.985358i	\(0.554539\pi\)
\(17\)			1060.62i			0.890101i	0.895506	+	0.445050i	\(0.146814\pi\)
							−0.895506	+	0.445050i	\(0.853186\pi\)
\(19\)	233.441	−	96.6945i	0.148352	−	0.0614494i	−0.307272	−	0.951622i	\(-0.599416\pi\)
							0.455624	+	0.890172i	\(0.349416\pi\)
\(23\)	−1550.81	+	1550.81i	−0.611277	+	0.611277i	−0.943279	−	0.332002i	\(-0.892276\pi\)
							0.332002	+	0.943279i	\(0.392276\pi\)
\(29\)	−1512.91	−	3652.50i	−0.334056	−	0.806482i	−0.998262	−	0.0589329i	\(-0.981230\pi\)
							0.664206	−	0.747550i	\(-0.268770\pi\)
\(31\)	1245.98			0.232866			0.116433	−	0.993199i	\(-0.462854\pi\)
							0.116433	+	0.993199i	\(0.462854\pi\)
\(37\)	7711.55	+	3194.23i	0.926056	+	0.383585i	0.794181	−	0.607682i	\(-0.207900\pi\)
							0.131875	+	0.991266i	\(0.457900\pi\)
\(41\)	472.797	−	472.797i	0.0439253	−	0.0439253i	−0.684803	−	0.728728i	\(-0.740112\pi\)
							0.728728	+	0.684803i	\(0.240112\pi\)
\(43\)	5.72744	−	13.8273i	0.000472378	−	0.00114042i	−0.923643	−	0.383254i	\(-0.874804\pi\)
							0.924116	+	0.382113i	\(0.124804\pi\)
\(47\)		−	9343.01i		−	0.616939i	−0.951234	−	0.308469i	\(-0.900183\pi\)
							0.951234	−	0.308469i	\(-0.0998167\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.6.g.a.5.11	✓	76
4.3	odd	2		128.6.g.a.113.7		76
32.13	even	8	inner	32.6.g.a.13.11	yes	76
32.19	odd	8		128.6.g.a.17.7		76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.6.g.a.5.11	✓	76	1.1	even	1	trivial
32.6.g.a.13.11	yes	76	32.13	even	8	inner
128.6.g.a.17.7		76	32.19	odd	8
128.6.g.a.113.7		76	4.3	odd	2