Embedded newform 429.2.bg.b.16.6

• Label: 429.2.bg.b.16.6
• 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			16.6
Character	\(\chi\)	\(=\)	429.16
Dual form			429.2.bg.b.295.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.471127 + 0.523240i) q^{2} +(-0.104528 + 0.994522i) q^{3} +(0.157238 + 1.49602i) q^{4} +(-0.263224 + 0.810122i) q^{5} +(-0.471127 - 0.523240i) q^{6} +(0.496767 + 4.72642i) q^{7} +(-1.99610 - 1.45025i) q^{8} +(-0.978148 - 0.207912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 14 q^{3} + 8 q^{4} - 6 q^{5} - 24 q^{8} + 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.471127	+	0.523240i	−0.333137	+	0.369987i	−0.886320	−	0.463074i	\(-0.846747\pi\)
							0.553182	+	0.833060i	\(0.313413\pi\)
\(3\)	−0.104528	+	0.994522i	−0.0603495	+	0.574187i
							\(5\)	−0.263224	+	0.810122i	−0.117718	+	0.362297i	−0.992504	−	0.122211i	\(-0.961002\pi\)
0.874787	+	0.484508i	\(0.161002\pi\)
\(7\)	0.496767	+	4.72642i	0.187760	+	1.78642i	0.531201	+	0.847246i	\(0.321741\pi\)
							−0.343441	+	0.939174i	\(0.611592\pi\)
\(11\)	3.11953	+	1.12628i	0.940574	+	0.339588i
							\(13\)	1.79396	−	3.12757i	0.497554	−	0.867433i
\(17\)	−2.69683	−	2.99513i	−0.654078	−	0.726427i								0.321298	−	0.946978i	\(-0.395881\pi\)
							−0.975375	+	0.220551i	\(0.929214\pi\)
\(19\)	−1.62654	+	0.724183i	−0.373154	+	0.166139i	−0.584741	−	0.811220i	\(-0.698804\pi\)
							0.211586	+	0.977359i	\(0.432137\pi\)
\(23\)	−0.814836	−	1.41134i	−0.169905	−	0.294284i	0.768481	−	0.639872i	\(-0.221013\pi\)
							−0.938386	+	0.345588i	\(0.887679\pi\)
\(29\)	3.24647	+	1.44542i	0.602855	+	0.268408i	0.685386	−	0.728180i	\(-0.259634\pi\)
							−0.0825310	+	0.996588i	\(0.526300\pi\)
\(31\)	0.330508	+	1.01720i	0.0593610	+	0.182694i	0.976340	−	0.216241i	\(-0.0693797\pi\)
							−0.916979	+	0.398936i	\(0.869380\pi\)
\(37\)	−2.91993	−	1.30004i	−0.480034	−	0.213725i	0.152431	−	0.988314i	\(-0.451290\pi\)
							−0.632464	+	0.774589i	\(0.717957\pi\)
\(41\)	0.687828	−	6.54425i	0.107421	−	1.02204i	−0.799479	−	0.600694i	\(-0.794891\pi\)
							0.906900	−	0.421346i	\(-0.138442\pi\)
\(43\)	−4.42184	+	7.65885i	−0.674324	+	1.16796i	0.302342	+	0.953200i	\(0.402232\pi\)
							−0.976666	+	0.214764i	\(0.931102\pi\)
\(47\)	1.45784	+	1.05918i	0.212647	+	0.154497i	0.689011	−	0.724751i	\(-0.258045\pi\)
							−0.476363	+	0.879248i	\(0.658045\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bg.b.16.6	✓	112
11.9	even	5	inner	429.2.bg.b.328.9	yes	112
13.9	even	3	inner	429.2.bg.b.412.9	yes	112
143.9	even	15	inner	429.2.bg.b.295.6	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bg.b.16.6	✓	112	1.1	even	1	trivial
429.2.bg.b.295.6	yes	112	143.9	even	15	inner
429.2.bg.b.328.9	yes	112	11.9	even	5	inner
429.2.bg.b.412.9	yes	112	13.9	even	3	inner