Embedded newform 32.6.g.a.5.10

• Label: 32.6.g.a.5.10
• 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			5.10
Character	\(\chi\)	\(=\)	32.5
Dual form			32.6.g.a.13.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0731871 + 5.65638i) q^{2} +(9.04699 + 21.8414i) q^{3} +(-31.9893 + 0.827948i) q^{4} +(83.7809 + 34.7032i) q^{5} +(-122.881 + 52.7718i) q^{6} +(-34.8236 - 34.8236i) q^{7} +(-7.02439 - 180.883i) q^{8} +(-223.371 + 223.371i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0731871	+	5.65638i	0.0129378	+	0.999916i
							\(3\)	9.04699	+	21.8414i	0.580365	+	1.40113i	0.892483	+	0.451082i	\(0.148962\pi\)
−0.312118	+	0.950043i	\(0.601038\pi\)
\(5\)	83.7809	+	34.7032i	1.49872	+	0.620790i	0.973193	−	0.229988i	\(-0.0738687\pi\)
							0.525525	+	0.850778i	\(0.323869\pi\)
\(7\)	−34.8236	−	34.8236i	−0.268614	−	0.268614i	0.559928	−	0.828541i	\(-0.310829\pi\)
							−0.828541	+	0.559928i	\(0.810829\pi\)
\(11\)	241.591	−	583.253i	0.602005	−	1.45337i	−0.269509	−	0.962998i	\(-0.586862\pi\)
							0.871514	−	0.490370i	\(-0.163138\pi\)
\(13\)	−569.664	+	235.962i	−0.934889	+	0.387244i	−0.797531	−	0.603278i	\(-0.793861\pi\)
							−0.137358	+	0.990521i	\(0.543861\pi\)
\(17\)			887.185i			0.744546i	0.928123	+	0.372273i	\(0.121422\pi\)
							−0.928123	+	0.372273i	\(0.878578\pi\)
\(19\)	1128.92	−	467.613i	0.717428	−	0.297168i	0.00605343	−	0.999982i	\(-0.498073\pi\)
							0.711374	+	0.702813i	\(0.248073\pi\)
\(23\)	2106.39	−	2106.39i	0.830271	−	0.830271i	−0.157283	−	0.987554i	\(-0.550273\pi\)
							0.987554	+	0.157283i	\(0.0502734\pi\)
\(29\)	934.441	+	2255.94i	0.206327	+	0.498118i	0.992839	−	0.119457i	\(-0.0381153\pi\)
							−0.786512	+	0.617575i	\(0.788115\pi\)
\(31\)	−8560.47			−1.59990			−0.799951	−	0.600065i	\(-0.795141\pi\)
							−0.799951	+	0.600065i	\(0.795141\pi\)
\(37\)	−732.927	−	303.588i	−0.0880149	−	0.0364570i	0.338242	−	0.941059i	\(-0.390168\pi\)
							−0.426257	+	0.904602i	\(0.640168\pi\)
\(41\)	−7247.87	+	7247.87i	−0.673366	+	0.673366i	−0.958490	−	0.285125i	\(-0.907965\pi\)
							0.285125	+	0.958490i	\(0.407965\pi\)
\(43\)	−1698.50	+	4100.55i	−0.140086	+	0.338198i	−0.978316	−	0.207120i	\(-0.933591\pi\)
							0.838229	+	0.545318i	\(0.183591\pi\)
\(47\)		−	6121.67i		−	0.404227i	−0.979362	−	0.202113i	\(-0.935219\pi\)
							0.979362	−	0.202113i	\(-0.0647810\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.6.g.a.5.10	✓	76
4.3	odd	2		128.6.g.a.113.3		76
32.13	even	8	inner	32.6.g.a.13.10	yes	76
32.19	odd	8		128.6.g.a.17.3		76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.6.g.a.5.10	✓	76	1.1	even	1	trivial
32.6.g.a.13.10	yes	76	32.13	even	8	inner
128.6.g.a.17.3		76	32.19	odd	8
128.6.g.a.113.3		76	4.3	odd	2