Embedded newform 32.4.g.a.29.10

• Label: 32.4.g.a.29.10
• Level: $32$
• Weight: $4$
• Character: 32.29
• Analytic conductor: $1.888$
• Analytic rank: $0$
• Dimension: $44$
• 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			29.10
Character	\(\chi\)	\(=\)	32.29
Dual form			32.4.g.a.21.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.14555 - 1.84300i) q^{2} +(-9.18660 - 3.80522i) q^{3} +(1.20673 - 7.90846i) q^{4} +(1.04223 + 2.51617i) q^{5} +(-26.7233 + 8.76661i) q^{6} +(16.1077 - 16.1077i) q^{7} +(-11.9862 - 19.1920i) q^{8} +(50.8221 + 50.8221i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{3}{8}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.14555	−	1.84300i	0.758565	−	0.651597i
							\(3\)	−9.18660	−	3.80522i	−1.76796	−	0.732314i	−0.995227	−	0.0975856i	\(-0.968888\pi\)
−0.772735	−	0.634728i	\(-0.781112\pi\)
\(5\)	1.04223	+	2.51617i	0.0932202	+	0.225053i	0.963611	−	0.267308i	\(-0.0861340\pi\)
							−0.870391	+	0.492361i	\(0.836134\pi\)
\(7\)	16.1077	−	16.1077i	0.869736	−	0.869736i	−0.122707	−	0.992443i	\(-0.539157\pi\)
							0.992443	+	0.122707i	\(0.0391575\pi\)
\(11\)	3.72541	−	1.54312i	0.102114	−	0.0422970i	−0.331042	−	0.943616i	\(-0.607400\pi\)
							0.433156	+	0.901319i	\(0.357400\pi\)
\(13\)	9.23975	−	22.3067i	0.197127	−	0.475906i	−0.794147	−	0.607726i	\(-0.792082\pi\)
							0.991274	+	0.131820i	\(0.0420820\pi\)
\(17\)		−	4.95290i		−	0.0706621i	−0.999376	−	0.0353310i	\(-0.988751\pi\)
							0.999376	−	0.0353310i	\(-0.0112486\pi\)
\(19\)	−26.0224	+	62.8237i	−0.314208	+	0.758565i	0.685332	+	0.728231i	\(0.259657\pi\)
							−0.999540	+	0.0303344i	\(0.990343\pi\)
\(23\)	82.8361	+	82.8361i	0.750979	+	0.750979i	0.974662	−	0.223683i	\(-0.0718079\pi\)
							−0.223683	+	0.974662i	\(0.571808\pi\)
\(29\)	150.525	+	62.3496i	0.963856	+	0.399242i	0.808422	−	0.588604i	\(-0.200322\pi\)
							0.155435	+	0.987846i	\(0.450322\pi\)
\(31\)	−141.577			−0.820258			−0.410129	−	0.912027i	\(-0.634516\pi\)
							−0.410129	+	0.912027i	\(0.634516\pi\)
\(37\)	−1.05672	−	2.55114i	−0.00469523	−	0.0113353i	0.921515	−	0.388344i	\(-0.126953\pi\)
							−0.926210	+	0.377009i	\(0.876953\pi\)
\(41\)	8.70340	+	8.70340i	0.0331523	+	0.0331523i	0.723489	−	0.690336i	\(-0.242537\pi\)
							−0.690336	+	0.723489i	\(0.742537\pi\)
\(43\)	290.048	−	120.142i	1.02865	−	0.426081i	0.196425	−	0.980519i	\(-0.437067\pi\)
							0.832225	+	0.554438i	\(0.187067\pi\)
\(47\)			450.158i			1.39707i	0.715576	+	0.698535i	\(0.246164\pi\)
							−0.715576	+	0.698535i	\(0.753836\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.4.g.a.29.10	yes	44
4.3	odd	2		128.4.g.a.81.11		44
8.3	odd	2		256.4.g.a.161.1		44
8.5	even	2		256.4.g.b.161.11		44
32.5	even	8		256.4.g.b.97.11		44
32.11	odd	8		128.4.g.a.49.11		44
32.21	even	8	inner	32.4.g.a.21.10	✓	44
32.27	odd	8		256.4.g.a.97.1		44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.4.g.a.21.10	✓	44	32.21	even	8	inner
32.4.g.a.29.10	yes	44	1.1	even	1	trivial
128.4.g.a.49.11		44	32.11	odd	8
128.4.g.a.81.11		44	4.3	odd	2
256.4.g.a.97.1		44	32.27	odd	8
256.4.g.a.161.1		44	8.3	odd	2
256.4.g.b.97.11		44	32.5	even	8
256.4.g.b.161.11		44	8.5	even	2