Embedded newform 429.2.p.b.230.16

• Label: 429.2.p.b.230.16
• Level: $429$
• Weight: $2$
• Character: 429.230
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(48\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			230.16
Character	\(\chi\)	\(=\)	429.230
Dual form			429.2.p.b.263.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.642823 + 1.11340i) q^{2} +(0.662940 + 1.60016i) q^{3} +(0.173557 + 0.300610i) q^{4} +1.88239i q^{5} +(-2.20777 - 0.290500i) q^{6} +(-0.680391 + 0.392824i) q^{7} -3.01756 q^{8} +(-2.12102 + 2.12162i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 2 q^{3} - 60 q^{4} - 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.642823	+	1.11340i	−0.454544	+	0.787294i	−0.998662	−	0.0517150i	\(-0.983531\pi\)
							0.544117	+	0.839009i	\(0.316865\pi\)
\(3\)	0.662940	+	1.60016i	0.382749	+	0.923852i
							\(5\)			1.88239i			0.841831i	0.907100	+	0.420916i	\(0.138291\pi\)
−0.907100	+	0.420916i	\(0.861709\pi\)
\(7\)	−0.680391	+	0.392824i	−0.257164	+	0.148473i	−0.623040	−	0.782190i	\(-0.714103\pi\)
							0.365876	+	0.930663i	\(0.380769\pi\)
\(11\)	−2.52484	−	2.15062i	−0.761268	−	0.648437i
							\(13\)	0.275603	−	3.59500i	0.0764385	−	0.997074i
\(17\)	2.60948	+	4.51975i	0.632891	+	1.09620i								0.986958	+	0.160979i	\(0.0514653\pi\)
							−0.354067	+	0.935220i	\(0.615201\pi\)
\(19\)	1.34814	−	0.778347i	0.309284	−	0.178565i	−0.337322	−	0.941389i	\(-0.609521\pi\)
							0.646606	+	0.762824i	\(0.276188\pi\)
\(23\)	4.08474	+	2.35833i	0.851727	+	0.491745i	0.861233	−	0.508210i	\(-0.169693\pi\)
							−0.00950587	+	0.999955i	\(0.503026\pi\)
\(29\)	−4.75033	+	8.22781i	−0.882114	+	1.52787i	−0.0331271	+	0.999451i	\(0.510547\pi\)
							−0.848987	+	0.528414i	\(0.822787\pi\)
\(31\)	7.47016			1.34168			0.670840	−	0.741602i	\(-0.265934\pi\)
							0.670840	+	0.741602i	\(0.265934\pi\)
\(37\)	−1.51663	+	2.62688i	−0.249333	+	0.431857i	−0.963341	−	0.268281i	\(-0.913545\pi\)
							0.714008	+	0.700137i	\(0.246878\pi\)
\(41\)	−0.941201	+	1.63021i	−0.146991	+	0.254596i	−0.930114	−	0.367271i	\(-0.880292\pi\)
							0.783123	+	0.621867i	\(0.213625\pi\)
\(43\)	−1.13181	+	0.653451i	−0.172600	+	0.0996504i	−0.583811	−	0.811890i	\(-0.698439\pi\)
							0.411212	+	0.911540i	\(0.365106\pi\)
\(47\)		−	2.92720i		−	0.426976i	−0.976946	−	0.213488i	\(-0.931518\pi\)
							0.976946	−	0.213488i	\(-0.0684825\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.p.b.230.16	yes	96
3.2	odd	2	inner	429.2.p.b.230.33	yes	96
11.10	odd	2	inner	429.2.p.b.230.34	yes	96
13.3	even	3	inner	429.2.p.b.263.15	yes	96
33.32	even	2	inner	429.2.p.b.230.15	✓	96
39.29	odd	6	inner	429.2.p.b.263.34	yes	96
143.120	odd	6	inner	429.2.p.b.263.33	yes	96
429.263	even	6	inner	429.2.p.b.263.16	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.p.b.230.15	✓	96	33.32	even	2	inner
429.2.p.b.230.16	yes	96	1.1	even	1	trivial
429.2.p.b.230.33	yes	96	3.2	odd	2	inner
429.2.p.b.230.34	yes	96	11.10	odd	2	inner
429.2.p.b.263.15	yes	96	13.3	even	3	inner
429.2.p.b.263.16	yes	96	429.263	even	6	inner
429.2.p.b.263.33	yes	96	143.120	odd	6	inner
429.2.p.b.263.34	yes	96	39.29	odd	6	inner