Embedded newform 429.2.bg.b.16.4

• Label: 429.2.bg.b.16.4
• 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			16.4
Character	\(\chi\)	\(=\)	429.16
Dual form			429.2.bg.b.295.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.12855 + 1.25338i) q^{2} +(-0.104528 + 0.994522i) q^{3} +(-0.0882831 - 0.839957i) q^{4} +(0.636771 - 1.95978i) q^{5} +(-1.12855 - 1.25338i) q^{6} +(-0.222872 - 2.12049i) q^{7} +(-1.57654 - 1.14542i) q^{8} +(-0.978148 - 0.207912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 14 q^{3} + 8 q^{4} - 6 q^{5} - 24 q^{8} + 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.12855	+	1.25338i	−0.798004	+	0.886274i	−0.995571	−	0.0940159i	\(-0.970030\pi\)
							0.197566	+	0.980290i	\(0.436696\pi\)
\(3\)	−0.104528	+	0.994522i	−0.0603495	+	0.574187i
							\(5\)	0.636771	−	1.95978i	0.284772	−	0.876439i	−0.701694	−	0.712478i	\(-0.747573\pi\)
0.986467	−	0.163961i	\(-0.0524272\pi\)
\(7\)	−0.222872	−	2.12049i	−0.0842378	−	0.801470i	−0.952331	−	0.305068i	\(-0.901321\pi\)
							0.868093	−	0.496402i	\(-0.165346\pi\)
\(11\)	3.07969	+	1.23106i	0.928562	+	0.371178i
							\(13\)	2.50603	+	2.59226i	0.695048	+	0.718963i
\(17\)	−0.358815	−	0.398504i	−0.0870253	−	0.0966514i								0.698059	−	0.716040i	\(-0.254047\pi\)
							−0.785085	+	0.619389i	\(0.787381\pi\)
\(19\)	6.15666	−	2.74112i	1.41243	−	0.628856i	0.448206	−	0.893930i	\(-0.352063\pi\)
							0.964228	+	0.265074i	\(0.0853965\pi\)
\(23\)	0.573160	+	0.992742i	0.119512	+	0.207001i	0.919574	−	0.392916i	\(-0.128534\pi\)
							−0.800062	+	0.599917i	\(0.795200\pi\)
\(29\)	−5.99798	−	2.67047i	−1.11380	−	0.495894i	−0.234475	−	0.972122i	\(-0.575337\pi\)
							−0.879322	+	0.476228i	\(0.842004\pi\)
\(31\)	1.16300	+	3.57936i	0.208882	+	0.642872i	0.999532	+	0.0306035i	\(0.00974290\pi\)
							−0.790650	+	0.612269i	\(0.790257\pi\)
\(37\)	7.24151	+	3.22413i	1.19050	+	0.530043i	0.903790	−	0.427976i	\(-0.140773\pi\)
							0.286706	+	0.958019i	\(0.407440\pi\)
\(41\)	0.576039	−	5.48064i	0.0899621	−	0.855933i	−0.852749	−	0.522321i	\(-0.825066\pi\)
							0.942711	−	0.333611i	\(-0.108267\pi\)
\(43\)	−0.686997	+	1.18991i	−0.104766	+	0.181460i	−0.913643	−	0.406518i	\(-0.866743\pi\)
							0.808877	+	0.587979i	\(0.200076\pi\)
\(47\)	0.859672	+	0.624589i	0.125396	+	0.0911056i	0.648716	−	0.761031i	\(-0.275307\pi\)
							−0.523319	+	0.852137i	\(0.675307\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bg.b.16.4	✓	112
11.9	even	5	inner	429.2.bg.b.328.11	yes	112
13.9	even	3	inner	429.2.bg.b.412.11	yes	112
143.9	even	15	inner	429.2.bg.b.295.4	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bg.b.16.4	✓	112	1.1	even	1	trivial
429.2.bg.b.295.4	yes	112	143.9	even	15	inner
429.2.bg.b.328.11	yes	112	11.9	even	5	inner
429.2.bg.b.412.11	yes	112	13.9	even	3	inner