Embedded newform 429.2.bg.b.16.2

• Label: 429.2.bg.b.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.b.295.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.53199 + 1.70145i) q^{2} +(-0.104528 + 0.994522i) q^{3} +(-0.338874 - 3.22417i) q^{4} +(0.485300 - 1.49360i) q^{5} +(-1.53199 - 1.70145i) q^{6} +(0.0971269 + 0.924101i) q^{7} +(2.30039 + 1.67133i) q^{8} +(-0.978148 - 0.207912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 14 q^{3} + 8 q^{4} - 6 q^{5} - 24 q^{8} + 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.53199	+	1.70145i	−1.08328	+	1.20311i	−0.105295	+	0.994441i	\(0.533579\pi\)
							−0.977987	+	0.208666i	\(0.933088\pi\)
\(3\)	−0.104528	+	0.994522i	−0.0603495	+	0.574187i
							\(5\)	0.485300	−	1.49360i	0.217033	−	0.667959i	−0.781970	−	0.623316i	\(-0.785785\pi\)
0.999003	−	0.0446426i	\(-0.0142149\pi\)
\(7\)	0.0971269	+	0.924101i	0.0367105	+	0.349277i	0.997424	+	0.0717344i	\(0.0228534\pi\)
							−0.960713	+	0.277543i	\(0.910480\pi\)
\(11\)	−2.74774	−	1.85740i	−0.828475	−	0.560026i
							\(13\)	−0.443667	−	3.57815i	−0.123051	−	0.992400i
\(17\)	−0.586178	−	0.651016i	−0.142169	−	0.157895i								0.667855	−	0.744291i	\(-0.267213\pi\)
							−0.810024	+	0.586397i	\(0.800546\pi\)
\(19\)	7.29201	−	3.24661i	1.67290	−	0.744824i	0.672916	−	0.739719i	\(-0.265042\pi\)
							0.999987	−	0.00510506i	\(-0.00162500\pi\)
\(23\)	−4.43756	−	7.68608i	−0.925295	−	1.60266i	−0.791086	−	0.611705i	\(-0.790484\pi\)
							−0.134209	−	0.990953i	\(-0.542849\pi\)
\(29\)	7.55856	+	3.36529i	1.40359	+	0.624918i	0.962187	−	0.272391i	\(-0.0878143\pi\)
							0.441403	+	0.897309i	\(0.354481\pi\)
\(31\)	−1.41164	−	4.34459i	−0.253539	−	0.780312i	−0.994114	−	0.108339i	\(-0.965447\pi\)
							0.740575	−	0.671973i	\(-0.234553\pi\)
\(37\)	−7.64759	−	3.40493i	−1.25726	−	0.559767i	−0.333501	−	0.942750i	\(-0.608230\pi\)
							−0.923755	+	0.382983i	\(0.874897\pi\)
\(41\)	−0.771161	+	7.33710i	−0.120435	+	1.14586i	0.752693	+	0.658372i	\(0.228755\pi\)
							−0.873128	+	0.487491i	\(0.837912\pi\)
\(43\)	−1.60518	+	2.78025i	−0.244787	+	0.423984i	−0.962072	−	0.272796i	\(-0.912051\pi\)
							0.717284	+	0.696780i	\(0.245385\pi\)
\(47\)	−2.00041	−	1.45339i	−0.291790	−	0.211998i	0.432253	−	0.901752i	\(-0.357719\pi\)
							−0.724044	+	0.689754i	\(0.757719\pi\)

(See \(a_n\) instead)

Twists

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

    

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