Embedded newform 32.4.g.a.13.5

• Label: 32.4.g.a.13.5
• Level: $32$
• Weight: $4$
• Character: 32.13
• 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			13.5
Character	\(\chi\)	\(=\)	32.13
Dual form			32.4.g.a.5.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.719102 - 2.73549i) q^{2} +(-3.21198 + 7.75440i) q^{3} +(-6.96578 + 3.93419i) q^{4} +(-13.6472 + 5.65283i) q^{5} +(23.5218 + 3.21012i) q^{6} +(9.07689 - 9.07689i) q^{7} +(15.7710 + 16.2257i) q^{8} +(-30.7220 - 30.7220i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{7}{8}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.719102	−	2.73549i	−0.254241	−	0.967141i
							\(3\)	−3.21198	+	7.75440i	−0.618145	+	1.49233i	0.235710	+	0.971824i	\(0.424259\pi\)
−0.853855	+	0.520511i	\(0.825741\pi\)
\(5\)	−13.6472	+	5.65283i	−1.22064	+	0.505605i	−0.897613	−	0.440785i	\(-0.854700\pi\)
							−0.323026	+	0.946390i	\(0.604700\pi\)
\(7\)	9.07689	−	9.07689i	0.490106	−	0.490106i	−0.418233	−	0.908340i	\(-0.637351\pi\)
							0.908340	+	0.418233i	\(0.137351\pi\)
\(11\)	16.7195	+	40.3646i	0.458285	+	1.10640i	0.969091	+	0.246702i	\(0.0793468\pi\)
							−0.510807	+	0.859696i	\(0.670653\pi\)
\(13\)	−38.4266	−	15.9168i	−0.819817	−	0.339580i	−0.0669538	−	0.997756i	\(-0.521328\pi\)
							−0.752864	+	0.658177i	\(0.771328\pi\)
\(17\)			92.5038i			1.31973i	0.751383	+	0.659867i	\(0.229387\pi\)
							−0.751383	+	0.659867i	\(0.770613\pi\)
\(19\)	−16.6799	−	6.90905i	−0.201402	−	0.0834235i	0.279702	−	0.960087i	\(-0.409764\pi\)
							−0.481104	+	0.876663i	\(0.659764\pi\)
\(23\)	95.6008	+	95.6008i	0.866702	+	0.866702i	0.992106	−	0.125404i	\(-0.0400228\pi\)
							−0.125404	+	0.992106i	\(0.540023\pi\)
\(29\)	19.3621	−	46.7443i	0.123981	−	0.299317i	−0.849687	−	0.527288i	\(-0.823209\pi\)
							0.973668	+	0.227971i	\(0.0732090\pi\)
\(31\)	−38.1477			−0.221017			−0.110508	−	0.993875i	\(-0.535248\pi\)
							−0.110508	+	0.993875i	\(0.535248\pi\)
\(37\)	227.529	−	94.2456i	1.01096	−	0.418754i	0.185156	−	0.982709i	\(-0.440721\pi\)
							0.825805	+	0.563955i	\(0.190721\pi\)
\(41\)	−279.334	−	279.334i	−1.06401	−	1.06401i	−0.997806	−	0.0662081i	\(-0.978910\pi\)
							−0.0662081	−	0.997806i	\(-0.521090\pi\)
\(43\)	112.235	+	270.960i	0.398039	+	0.960952i	0.988131	+	0.153616i	\(0.0490918\pi\)
							−0.590091	+	0.807337i	\(0.700908\pi\)
\(47\)			321.955i			0.999191i	0.866259	+	0.499596i	\(0.166518\pi\)
							−0.866259	+	0.499596i	\(0.833482\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.4.g.a.13.5	yes	44
4.3	odd	2		128.4.g.a.17.10		44
8.3	odd	2		256.4.g.a.33.2		44
8.5	even	2		256.4.g.b.33.10		44
32.5	even	8	inner	32.4.g.a.5.5	✓	44
32.11	odd	8		256.4.g.a.225.2		44
32.21	even	8		256.4.g.b.225.10		44
32.27	odd	8		128.4.g.a.113.10		44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.4.g.a.5.5	✓	44	32.5	even	8	inner
32.4.g.a.13.5	yes	44	1.1	even	1	trivial
128.4.g.a.17.10		44	4.3	odd	2
128.4.g.a.113.10		44	32.27	odd	8
256.4.g.a.33.2		44	8.3	odd	2
256.4.g.a.225.2		44	32.11	odd	8
256.4.g.b.33.10		44	8.5	even	2
256.4.g.b.225.10		44	32.21	even	8