Embedded newform 429.2.bg.a.16.10

• Label: 429.2.bg.a.16.10
• Level: $429$
• Weight: $2$
• Character: 429.16
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.bg (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			16.10
Character	\(\chi\)	\(=\)	429.16
Dual form			429.2.bg.a.295.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.844998 - 0.938465i) q^{2} +(0.104528 - 0.994522i) q^{3} +(0.0423614 + 0.403042i) q^{4} +(1.13050 - 3.47932i) q^{5} +(-0.844998 - 0.938465i) q^{6} +(-0.100820 - 0.959240i) q^{7} +(2.45734 + 1.78536i) q^{8} +(-0.978148 - 0.207912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 14 q^{3} + 20 q^{4} + 6 q^{5} + 24 q^{8} + 14 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.844998	−	0.938465i	0.597504	−	0.663595i	−0.366210	−	0.930532i	\(-0.619345\pi\)
							0.963714	+	0.266937i	\(0.0860116\pi\)
\(3\)	0.104528	−	0.994522i	0.0603495	−	0.574187i
							\(5\)	1.13050	−	3.47932i	0.505575	−	1.55600i	−0.294228	−	0.955735i	\(-0.595062\pi\)
0.799802	−	0.600263i	\(-0.204938\pi\)
\(7\)	−0.100820	−	0.959240i	−0.0381064	−	0.362558i	−0.996914	−	0.0785019i	\(-0.974986\pi\)
							0.958808	−	0.284057i	\(-0.0916803\pi\)
\(11\)	−2.92792	+	1.55798i	−0.882801	+	0.469747i
							\(13\)	−0.198657	−	3.60007i	−0.0550976	−	0.998481i
\(17\)	1.41916	+	1.57614i	0.344197	+	0.382269i								0.890243	−	0.455486i	\(-0.150534\pi\)
							−0.546046	+	0.837755i	\(0.683868\pi\)
\(19\)	1.12592	−	0.501292i	0.258304	−	0.115004i	−0.273500	−	0.961872i	\(-0.588181\pi\)
							0.531803	+	0.846868i	\(0.321515\pi\)
\(23\)	4.21989	+	7.30906i	0.879907	+	1.52404i	0.851442	+	0.524449i	\(0.175729\pi\)
							0.0284655	+	0.999595i	\(0.490938\pi\)
\(29\)	−2.94341	−	1.31049i	−0.546578	−	0.243352i	0.114822	−	0.993386i	\(-0.463370\pi\)
							−0.661399	+	0.750034i	\(0.730037\pi\)
\(31\)	−2.72490	−	8.38638i	−0.489406	−	1.50624i	−0.825496	−	0.564408i	\(-0.809105\pi\)
							0.336090	−	0.941830i	\(-0.390895\pi\)
\(37\)	1.79828	+	0.800646i	0.295636	+	0.131625i	0.549195	−	0.835694i	\(-0.314935\pi\)
							−0.253559	+	0.967320i	\(0.581601\pi\)
\(41\)	0.118834	−	1.13063i	0.0185587	−	0.176574i	−0.981316	−	0.192403i	\(-0.938372\pi\)
							0.999875	+	0.0158287i	\(0.00503865\pi\)
\(43\)	−3.61068	+	6.25387i	−0.550623	+	0.953707i	0.447607	+	0.894230i	\(0.352276\pi\)
							−0.998230	+	0.0594763i	\(0.981057\pi\)
\(47\)	8.77669	+	6.37664i	1.28021	+	0.930128i	0.999559	−	0.0296818i	\(-0.00944940\pi\)
							0.280652	+	0.959810i	\(0.409449\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bg.a.16.10	✓	112
11.9	even	5	inner	429.2.bg.a.328.5	yes	112
13.9	even	3	inner	429.2.bg.a.412.5	yes	112
143.9	even	15	inner	429.2.bg.a.295.10	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bg.a.16.10	✓	112	1.1	even	1	trivial
429.2.bg.a.295.10	yes	112	143.9	even	15	inner
429.2.bg.a.328.5	yes	112	11.9	even	5	inner
429.2.bg.a.412.5	yes	112	13.9	even	3	inner