Embedded newform 429.2.bg.a.16.2

• Label: 429.2.bg.a.16.2
• 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			16.2
Character	\(\chi\)	\(=\)	429.16
Dual form			429.2.bg.a.295.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.68093 + 1.86686i) q^{2} +(0.104528 - 0.994522i) q^{3} +(-0.450589 - 4.28707i) q^{4} +(1.24954 - 3.84568i) q^{5} +(1.68093 + 1.86686i) q^{6} +(0.0277297 + 0.263830i) q^{7} +(4.69608 + 3.41190i) q^{8} +(-0.978148 - 0.207912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 14 q^{3} + 20 q^{4} + 6 q^{5} + 24 q^{8} + 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.68093	+	1.86686i	−1.18860	+	1.32007i	−0.252803	+	0.967518i	\(0.581352\pi\)
							−0.935793	+	0.352551i	\(0.885314\pi\)
\(3\)	0.104528	−	0.994522i	0.0603495	−	0.574187i
							\(5\)	1.24954	−	3.84568i	0.558811	−	1.71984i	−0.126851	−	0.991922i	\(-0.540487\pi\)
0.685662	−	0.727920i	\(-0.259513\pi\)
\(7\)	0.0277297	+	0.263830i	0.0104808	+	0.0997185i	0.998512	−	0.0545378i	\(-0.0173685\pi\)
							−0.988031	+	0.154256i	\(0.950702\pi\)
\(11\)	−0.565933	−	3.26798i	−0.170635	−	0.985334i
							\(13\)	−2.48558	+	2.61187i	−0.689377	+	0.724403i
\(17\)	1.18585	+	1.31702i	0.287610	+	0.319423i								0.869585	−	0.493783i	\(-0.164386\pi\)
							−0.581975	+	0.813207i	\(0.697720\pi\)
\(19\)	−1.20356	+	0.535859i	−0.276115	+	0.122935i	−0.540123	−	0.841586i	\(-0.681622\pi\)
							0.264008	+	0.964521i	\(0.414956\pi\)
\(23\)	−3.03803	−	5.26202i	−0.633473	−	1.09721i	−0.986836	−	0.161722i	\(-0.948295\pi\)
							0.353363	−	0.935486i	\(-0.385038\pi\)
\(29\)	−7.73423	−	3.44350i	−1.43621	−	0.639442i	−0.466684	−	0.884424i	\(-0.654551\pi\)
							−0.969528	+	0.244982i	\(0.921218\pi\)
\(31\)	1.49980	+	4.61590i	0.269371	+	0.829039i	0.990654	+	0.136399i	\(0.0435528\pi\)
							−0.721283	+	0.692641i	\(0.756447\pi\)
\(37\)	5.17903	+	2.30585i	0.851427	+	0.379080i	0.785588	−	0.618750i	\(-0.212361\pi\)
							0.0658396	+	0.997830i	\(0.479027\pi\)
\(41\)	0.745713	−	7.09499i	0.116461	−	1.10805i	−0.767681	−	0.640832i	\(-0.778589\pi\)
							0.884142	−	0.467219i	\(-0.154744\pi\)
\(43\)	−2.10913	+	3.65312i	−0.321640	+	0.557096i	−0.980827	−	0.194883i	\(-0.937567\pi\)
							0.659187	+	0.751979i	\(0.270901\pi\)
\(47\)	8.36232	+	6.07558i	1.21977	+	0.886215i	0.996081	−	0.0884459i	\(-0.0281900\pi\)
							0.223689	+	0.974661i	\(0.428190\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bg.a.16.2	✓	112
11.9	even	5	inner	429.2.bg.a.328.13	yes	112
13.9	even	3	inner	429.2.bg.a.412.13	yes	112
143.9	even	15	inner	429.2.bg.a.295.2	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bg.a.16.2	✓	112	1.1	even	1	trivial
429.2.bg.a.295.2	yes	112	143.9	even	15	inner
429.2.bg.a.328.13	yes	112	11.9	even	5	inner
429.2.bg.a.412.13	yes	112	13.9	even	3	inner