Embedded newform 32.6.g.a.5.3

• Label: 32.6.g.a.5.3
• 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			5.3
Character	\(\chi\)	\(=\)	32.5
Dual form			32.6.g.a.13.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.36185 - 1.80293i) q^{2} +(0.399918 + 0.965487i) q^{3} +(25.4989 + 19.3341i) q^{4} +(-41.3553 - 17.1299i) q^{5} +(-0.403590 - 5.89782i) q^{6} +(125.552 + 125.552i) q^{7} +(-101.863 - 149.639i) q^{8} +(171.055 - 171.055i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.36185	−	1.80293i	−0.947850	−	0.318717i
							\(3\)	0.399918	+	0.965487i	0.0256547	+	0.0619360i	0.936188	−	0.351500i	\(-0.114328\pi\)
−0.910533	+	0.413436i	\(0.864328\pi\)
\(5\)	−41.3553	−	17.1299i	−0.739786	−	0.306429i	−0.0192193	−	0.999815i	\(-0.506118\pi\)
							−0.720566	+	0.693386i	\(0.756118\pi\)
\(7\)	125.552	+	125.552i	0.968450	+	0.968450i	0.999517	−	0.0310670i	\(-0.00989052\pi\)
							−0.0310670	+	0.999517i	\(0.509891\pi\)
\(11\)	264.380	−	638.269i	0.658789	−	1.59046i	−0.140887	−	0.990026i	\(-0.544995\pi\)
							0.799676	−	0.600432i	\(-0.205005\pi\)
\(13\)	686.551	−	284.379i	1.12672	−	0.466701i	0.260052	−	0.965595i	\(-0.416260\pi\)
							0.866663	+	0.498894i	\(0.166260\pi\)
\(17\)			1748.22i			1.46714i	0.679611	+	0.733572i	\(0.262148\pi\)
							−0.679611	+	0.733572i	\(0.737852\pi\)
\(19\)	−280.732	+	116.283i	−0.178406	+	0.0738980i	−0.470098	−	0.882614i	\(-0.655782\pi\)
							0.291693	+	0.956512i	\(0.405782\pi\)
\(23\)	2378.63	−	2378.63i	0.937577	−	0.937577i	−0.0605860	−	0.998163i	\(-0.519297\pi\)
							0.998163	+	0.0605860i	\(0.0192969\pi\)
\(29\)	1476.83	+	3565.37i	0.326088	+	0.787245i	0.998876	+	0.0474097i	\(0.0150966\pi\)
							−0.672788	+	0.739835i	\(0.734903\pi\)
\(31\)	−2208.06			−0.412674			−0.206337	−	0.978481i	\(-0.566154\pi\)
							−0.206337	+	0.978481i	\(0.566154\pi\)
\(37\)	5673.53	+	2350.05i	0.681317	+	0.282211i	0.696377	−	0.717676i	\(-0.254794\pi\)
							−0.0150604	+	0.999887i	\(0.504794\pi\)
\(41\)	−9188.67	+	9188.67i	−0.853676	+	0.853676i	−0.990584	−	0.136908i	\(-0.956284\pi\)
							0.136908	+	0.990584i	\(0.456284\pi\)
\(43\)	4699.03	−	11344.5i	0.387559	−	0.935649i	−0.602897	−	0.797819i	\(-0.705987\pi\)
							0.990456	−	0.137830i	\(-0.0440129\pi\)
\(47\)		−	7212.14i		−	0.476233i	−0.971237	−	0.238117i	\(-0.923470\pi\)
							0.971237	−	0.238117i	\(-0.0765300\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.6.g.a.5.3	✓	76
4.3	odd	2		128.6.g.a.113.9		76
32.13	even	8	inner	32.6.g.a.13.3	yes	76
32.19	odd	8		128.6.g.a.17.9		76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.6.g.a.5.3	✓	76	1.1	even	1	trivial
32.6.g.a.13.3	yes	76	32.13	even	8	inner
128.6.g.a.17.9		76	32.19	odd	8
128.6.g.a.113.9		76	4.3	odd	2