Embedded newform 429.2.bg.a.16.3

• Label: 429.2.bg.a.16.3
• 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			16.3
Character	\(\chi\)	\(=\)	429.16
Dual form			429.2.bg.a.295.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.46128 + 1.62292i) q^{2} +(0.104528 - 0.994522i) q^{3} +(-0.289460 - 2.75403i) q^{4} +(-0.299446 + 0.921600i) q^{5} +(1.46128 + 1.62292i) q^{6} +(-0.235463 - 2.24028i) q^{7} +(1.35900 + 0.987368i) q^{8} +(-0.978148 - 0.207912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 14 q^{3} + 20 q^{4} + 6 q^{5} + 24 q^{8} + 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.46128	+	1.62292i	−1.03328	+	1.14758i	−0.0443779	+	0.999015i	\(0.514131\pi\)
							−0.988903	+	0.148560i	\(0.952536\pi\)
\(3\)	0.104528	−	0.994522i	0.0603495	−	0.574187i
							\(5\)	−0.299446	+	0.921600i	−0.133916	+	0.412152i	−0.995420	−	0.0955996i	\(-0.969523\pi\)
0.861504	+	0.507751i	\(0.169523\pi\)
\(7\)	−0.235463	−	2.24028i	−0.0889967	−	0.846747i	−0.944403	−	0.328790i	\(-0.893359\pi\)
							0.855406	−	0.517958i	\(-0.173307\pi\)
\(11\)	−2.22438	+	2.46011i	−0.670676	+	0.741750i
							\(13\)	1.70341	+	3.17779i	0.472442	+	0.881362i
\(17\)	3.36960	+	3.74232i	0.817247	+	0.907645i								0.997104	−	0.0760479i	\(-0.0242302\pi\)
							−0.179857	+	0.983693i	\(0.557564\pi\)
\(19\)	−0.347496	+	0.154715i	−0.0797211	+	0.0354941i	−0.446210	−	0.894928i	\(-0.647227\pi\)
							0.366489	+	0.930422i	\(0.380560\pi\)
\(23\)	1.27091	+	2.20129i	0.265004	+	0.459000i	0.967565	−	0.252623i	\(-0.0812934\pi\)
							−0.702561	+	0.711624i	\(0.747960\pi\)
\(29\)	6.91247	+	3.07763i	1.28361	+	0.571502i	0.931257	−	0.364364i	\(-0.118714\pi\)
							0.352357	+	0.935866i	\(0.385380\pi\)
\(31\)	−0.0642090	−	0.197615i	−0.0115323	−	0.0354927i	0.945125	−	0.326710i	\(-0.105940\pi\)
							−0.956657	+	0.291217i	\(0.905940\pi\)
\(37\)	−1.62313	−	0.722663i	−0.266840	−	0.118805i	0.268955	−	0.963153i	\(-0.413322\pi\)
							−0.535796	+	0.844348i	\(0.679988\pi\)
\(41\)	−0.731782	+	6.96244i	−0.114285	+	1.08735i	0.775619	+	0.631202i	\(0.217438\pi\)
							−0.889904	+	0.456149i	\(0.849229\pi\)
\(43\)	−6.37455	+	11.0410i	−0.972110	+	1.68374i	−0.282950	+	0.959135i	\(0.591313\pi\)
							−0.689160	+	0.724609i	\(0.742020\pi\)
\(47\)	−3.93489	−	2.85887i	−0.573963	−	0.417008i	0.262580	−	0.964910i	\(-0.415427\pi\)
							−0.836543	+	0.547902i	\(0.815427\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bg.a.16.3	✓	112
11.9	even	5	inner	429.2.bg.a.328.12	yes	112
13.9	even	3	inner	429.2.bg.a.412.12	yes	112
143.9	even	15	inner	429.2.bg.a.295.3	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bg.a.16.3	✓	112	1.1	even	1	trivial
429.2.bg.a.295.3	yes	112	143.9	even	15	inner
429.2.bg.a.328.12	yes	112	11.9	even	5	inner
429.2.bg.a.412.12	yes	112	13.9	even	3	inner