Embedded newform 32.4.g.a.13.7

• Label: 32.4.g.a.13.7
• 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			13.7
Character	\(\chi\)	\(=\)	32.13
Dual form			32.4.g.a.5.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.17521 - 2.57272i) q^{2} +(0.729459 - 1.76107i) q^{3} +(-5.23776 - 6.04698i) q^{4} +(4.29822 - 1.78038i) q^{5} +(-3.67347 - 3.94632i) q^{6} +(1.47807 - 1.47807i) q^{7} +(-21.7126 + 6.36880i) q^{8} +(16.5226 + 16.5226i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.17521	−	2.57272i	0.415500	−	0.909593i
							\(3\)	0.729459	−	1.76107i	0.140384	−	0.338918i	−0.838013	−	0.545650i	\(-0.816283\pi\)
0.978398	+	0.206732i	\(0.0662828\pi\)
\(5\)	4.29822	−	1.78038i	0.384444	−	0.159242i	−0.182087	−	0.983282i	\(-0.558285\pi\)
							0.566531	+	0.824040i	\(0.308285\pi\)
\(7\)	1.47807	−	1.47807i	0.0798085	−	0.0798085i	−0.666076	−	0.745884i	\(-0.732027\pi\)
							0.745884	+	0.666076i	\(0.232027\pi\)
\(11\)	0.854325	+	2.06252i	0.0234172	+	0.0565340i	0.935156	−	0.354237i	\(-0.115259\pi\)
							−0.911738	+	0.410771i	\(0.865259\pi\)
\(13\)	40.9706	+	16.9706i	0.874092	+	0.362061i	0.774203	−	0.632938i	\(-0.218151\pi\)
							0.0998893	+	0.994999i	\(0.468151\pi\)
\(17\)			73.1063i			1.04299i	0.853253	+	0.521496i	\(0.174626\pi\)
							−0.853253	+	0.521496i	\(0.825374\pi\)
\(19\)	−18.2478	−	7.55849i	−0.220333	−	0.0912650i	0.269786	−	0.962920i	\(-0.413047\pi\)
							−0.490120	+	0.871655i	\(0.663047\pi\)
\(23\)	−144.221	−	144.221i	−1.30748	−	1.30748i	−0.923228	−	0.384253i	\(-0.874459\pi\)
							−0.384253	−	0.923228i	\(-0.625541\pi\)
\(29\)	80.7690	−	194.994i	0.517187	−	1.24860i	−0.422437	−	0.906392i	\(-0.638825\pi\)
							0.939624	−	0.342208i	\(-0.111175\pi\)
\(31\)	−168.830			−0.978153			−0.489077	−	0.872241i	\(-0.662666\pi\)
							−0.489077	+	0.872241i	\(0.662666\pi\)
\(37\)	−72.0103	+	29.8277i	−0.319958	+	0.132531i	−0.536881	−	0.843658i	\(-0.680398\pi\)
							0.216924	+	0.976189i	\(0.430398\pi\)
\(41\)	−141.297	−	141.297i	−0.538215	−	0.538215i	0.384789	−	0.923005i	\(-0.374274\pi\)
							−0.923005	+	0.384789i	\(0.874274\pi\)
\(43\)	161.344	+	389.519i	0.572203	+	1.38142i	0.899675	+	0.436560i	\(0.143803\pi\)
							−0.327472	+	0.944861i	\(0.606197\pi\)
\(47\)			239.015i			0.741787i	0.928675	+	0.370893i	\(0.120948\pi\)
							−0.928675	+	0.370893i	\(0.879052\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.4.g.a.13.7	yes	44
4.3	odd	2		128.4.g.a.17.5		44
8.3	odd	2		256.4.g.a.33.7		44
8.5	even	2		256.4.g.b.33.5		44
32.5	even	8	inner	32.4.g.a.5.7	✓	44
32.11	odd	8		256.4.g.a.225.7		44
32.21	even	8		256.4.g.b.225.5		44
32.27	odd	8		128.4.g.a.113.5		44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.4.g.a.5.7	✓	44	32.5	even	8	inner
32.4.g.a.13.7	yes	44	1.1	even	1	trivial
128.4.g.a.17.5		44	4.3	odd	2
128.4.g.a.113.5		44	32.27	odd	8
256.4.g.a.33.7		44	8.3	odd	2
256.4.g.a.225.7		44	32.11	odd	8
256.4.g.b.33.5		44	8.5	even	2
256.4.g.b.225.5		44	32.21	even	8