Embedded newform 429.2.bn.a.4.9

• Label: 429.2.bn.a.4.9
• Level: $429$
• Weight: $2$
• Character: 429.4
• 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.bn (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			4.9
Character	\(\chi\)	\(=\)	429.4
Dual form			429.2.bn.a.322.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.275867 + 0.619607i) q^{2} +(-0.669131 - 0.743145i) q^{3} +(1.03045 - 1.14443i) q^{4} +(-1.77061 - 2.43703i) q^{5} +(0.275867 - 0.619607i) q^{6} +(3.22654 + 2.90519i) q^{7} +(2.28346 + 0.741942i) q^{8} +(-0.104528 + 0.994522i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 14 q^{3} - 20 q^{4} + 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.275867	+	0.619607i	0.195067	+	0.438128i	0.984425	−	0.175805i	\(-0.0562529\pi\)
							−0.789358	+	0.613934i	\(0.789586\pi\)
\(3\)	−0.669131	−	0.743145i	−0.386323	−	0.429055i
							\(5\)	−1.77061	−	2.43703i	−0.791841	−	1.08987i	−0.993876	−	0.110497i	\(-0.964756\pi\)
0.202036	−	0.979378i	\(-0.435244\pi\)
\(7\)	3.22654	+	2.90519i	1.21952	+	1.09806i	0.992254	+	0.124227i	\(0.0396450\pi\)
							0.227265	+	0.973833i	\(0.427022\pi\)
\(11\)	−3.10204	−	1.17360i	−0.935301	−	0.353852i
							\(13\)	1.10633	−	3.43162i	0.306841	−	0.951761i
\(17\)	3.14904	+	1.40204i	0.763754	+	0.340045i								0.751373	−	0.659878i	\(-0.229392\pi\)
							0.0123816	+	0.999923i	\(0.496059\pi\)
\(19\)	−1.50514	−	7.08113i	−0.345303	−	1.62452i	−0.717632	−	0.696423i	\(-0.754774\pi\)
							0.372329	−	0.928101i	\(-0.378559\pi\)
\(23\)	1.85891	−	3.21973i	0.387610	−	0.671360i	−0.604518	−	0.796592i	\(-0.706634\pi\)
							0.992128	+	0.125232i	\(0.0399674\pi\)
\(29\)	−7.31237	−	1.55429i	−1.35787	−	0.288625i	−0.529267	−	0.848456i	\(-0.677533\pi\)
							−0.828607	+	0.559831i	\(0.810866\pi\)
\(31\)	0.863702	−	1.18878i	0.155125	−	0.213512i	−0.724380	−	0.689401i	\(-0.757874\pi\)
							0.879505	+	0.475889i	\(0.157874\pi\)
\(37\)	−1.64478	+	7.73809i	−0.270401	+	1.27213i	0.607895	+	0.794017i	\(0.292014\pi\)
							−0.878296	+	0.478118i	\(0.841319\pi\)
\(41\)	−2.80306	+	2.52388i	−0.437764	+	0.394164i	−0.858330	−	0.513097i	\(-0.828498\pi\)
							0.420567	+	0.907262i	\(0.361831\pi\)
\(43\)	4.82368	+	8.35485i	0.735604	+	1.27410i	0.954458	+	0.298345i	\(0.0964348\pi\)
							−0.218854	+	0.975758i	\(0.570232\pi\)
\(47\)	−1.10331	−	0.358488i	−0.160935	−	0.0522909i	0.227442	−	0.973792i	\(-0.426964\pi\)
							−0.388376	+	0.921501i	\(0.626964\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bn.a.4.9	✓	112
11.3	even	5	inner	429.2.bn.a.355.9	yes	112
13.10	even	6	inner	429.2.bn.a.400.9	yes	112
143.36	even	30	inner	429.2.bn.a.322.9	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bn.a.4.9	✓	112	1.1	even	1	trivial
429.2.bn.a.322.9	yes	112	143.36	even	30	inner
429.2.bn.a.355.9	yes	112	11.3	even	5	inner
429.2.bn.a.400.9	yes	112	13.10	even	6	inner