Embedded newform 429.2.bg.a.16.1

• Label: 429.2.bg.a.16.1
• 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			16.1
Character	\(\chi\)	\(=\)	429.16
Dual form			429.2.bg.a.295.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.80037 + 1.99951i) q^{2} +(0.104528 - 0.994522i) q^{3} +(-0.547661 - 5.21064i) q^{4} +(-0.759776 + 2.33835i) q^{5} +(1.80037 + 1.99951i) q^{6} +(0.0209204 + 0.199044i) q^{7} +(7.05123 + 5.12302i) q^{8} +(-0.978148 - 0.207912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 14 q^{3} + 20 q^{4} + 6 q^{5} + 24 q^{8} + 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.80037	+	1.99951i	−1.27305	+	1.41387i	−0.407374	+	0.913261i	\(0.633555\pi\)
							−0.865676	+	0.500604i	\(0.833111\pi\)
\(3\)	0.104528	−	0.994522i	0.0603495	−	0.574187i
							\(5\)	−0.759776	+	2.33835i	−0.339782	+	1.04574i	0.624536	+	0.780996i	\(0.285288\pi\)
−0.964318	+	0.264746i	\(0.914712\pi\)
\(7\)	0.0209204	+	0.199044i	0.00790716	+	0.0752316i	0.997764	−	0.0668379i	\(-0.0212910\pi\)
							−0.989857	+	0.142069i	\(0.954624\pi\)
\(11\)	2.91130	−	1.58882i	0.877789	−	0.479047i
							\(13\)	0.397855	−	3.58353i	0.110345	−	0.993893i
\(17\)	−4.10066	−	4.55425i	−0.994557	−	1.10457i								−0.994518	−	0.104567i	\(-0.966654\pi\)
							−3.88257e−5	−	1.00000i	\(-0.500012\pi\)
\(19\)	2.84705	−	1.26759i	0.653158	−	0.290805i	−0.0532797	−	0.998580i	\(-0.516968\pi\)
							0.706438	+	0.707775i	\(0.250301\pi\)
\(23\)	2.53472	+	4.39026i	0.528525	+	0.915433i	0.999447	+	0.0332577i	\(0.0105882\pi\)
							−0.470921	+	0.882175i	\(0.656078\pi\)
\(29\)	8.22656	+	3.66270i	1.52763	+	0.680146i	0.986942	−	0.161075i	\(-0.0514962\pi\)
							0.540691	+	0.841221i	\(0.318163\pi\)
\(31\)	2.45796	+	7.56483i	0.441463	+	1.35868i	0.886316	+	0.463080i	\(0.153256\pi\)
							−0.444853	+	0.895603i	\(0.646744\pi\)
\(37\)	4.32731	+	1.92664i	0.711405	+	0.316738i	0.730341	−	0.683083i	\(-0.239361\pi\)
							−0.0189359	+	0.999821i	\(0.506028\pi\)
\(41\)	0.0999179	−	0.950655i	0.0156046	−	0.148467i	−0.983945	−	0.178471i	\(-0.942885\pi\)
							0.999550	+	0.0300032i	\(0.00955174\pi\)
\(43\)	3.42964	−	5.94030i	0.523015	−	0.905888i	−0.476627	−	0.879106i	\(-0.658141\pi\)
							0.999641	−	0.0267820i	\(-0.00852600\pi\)
\(47\)	0.672741	+	0.488775i	0.0981294	+	0.0712952i	0.635768	−	0.771880i	\(-0.280684\pi\)
							−0.537639	+	0.843175i	\(0.680684\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bg.a.16.1	✓	112
11.9	even	5	inner	429.2.bg.a.328.14	yes	112
13.9	even	3	inner	429.2.bg.a.412.14	yes	112
143.9	even	15	inner	429.2.bg.a.295.1	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bg.a.16.1	✓	112	1.1	even	1	trivial
429.2.bg.a.295.1	yes	112	143.9	even	15	inner
429.2.bg.a.328.14	yes	112	11.9	even	5	inner
429.2.bg.a.412.14	yes	112	13.9	even	3	inner