Embedded newform 429.2.bs.b.46.9

• Label: 429.2.bs.b.46.9
• Level: $429$
• Weight: $2$
• Character: 429.46
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $224$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.bs (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(224\)
Relative dimension:	\(14\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			46.9
Character	\(\chi\)	\(=\)	429.46
Dual form			429.2.bs.b.28.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.14984 + 0.0602607i) q^{2} +(-0.104528 - 0.994522i) q^{3} +(-0.670538 - 0.0704764i) q^{4} +(-0.555848 + 1.09091i) q^{5} +(-0.0602607 - 1.14984i) q^{6} +(-3.71251 + 3.00633i) q^{7} +(-3.04125 - 0.481687i) q^{8} +(-0.978148 + 0.207912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 224 q + 28 q^{3} + 6 q^{5} + 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.14984	+	0.0602607i	0.813061	+	0.0426107i	0.454334	−	0.890832i	\(-0.349877\pi\)
							0.358728	+	0.933442i	\(0.383211\pi\)
\(3\)	−0.104528	−	0.994522i	−0.0603495	−	0.574187i
							\(5\)	−0.555848	+	1.09091i	−0.248583	+	0.487871i	−0.981256	−	0.192710i	\(-0.938272\pi\)
0.732673	+	0.680581i	\(0.238272\pi\)
\(7\)	−3.71251	+	3.00633i	−1.40320	+	1.13629i	−0.430936	+	0.902382i	\(0.641817\pi\)
							−0.972260	+	0.233903i	\(0.924850\pi\)
\(11\)	−2.85813	+	1.68259i	−0.861759	+	0.507319i
							\(13\)	1.18608	−	3.40488i	0.328959	−	0.944344i
\(17\)	2.10954	−	2.34289i	0.511640	−	0.568233i								−0.430869	−	0.902414i	\(-0.641793\pi\)
							0.942509	+	0.334181i	\(0.108460\pi\)
\(19\)	−4.85340	+	1.86304i	−1.11345	+	0.427412i	−0.844399	−	0.535715i	\(-0.820042\pi\)
							−0.269047	+	0.963127i	\(0.586709\pi\)
\(23\)	−1.49068	−	0.860647i	−0.310829	−	0.179457i	0.336468	−	0.941695i	\(-0.390768\pi\)
							−0.647297	+	0.762237i	\(0.724101\pi\)
\(29\)	0.786963	+	1.76755i	0.146135	+	0.328225i	0.971751	−	0.236007i	\(-0.0758388\pi\)
							−0.825616	+	0.564232i	\(0.809172\pi\)
\(31\)	4.40954	−	2.24677i	0.791977	−	0.403532i	−0.0107028	−	0.999943i	\(-0.503407\pi\)
							0.802680	+	0.596410i	\(0.203407\pi\)
\(37\)	−2.15390	+	5.61109i	−0.354098	+	0.922458i	0.634641	+	0.772807i	\(0.281148\pi\)
							−0.988739	+	0.149650i	\(0.952185\pi\)
\(41\)	−1.20761	−	0.977905i	−0.188597	−	0.152723i	0.530372	−	0.847765i	\(-0.322052\pi\)
							−0.718969	+	0.695042i	\(0.755386\pi\)
\(43\)	−2.90194	−	5.02630i	−0.442541	−	0.766504i	0.555336	−	0.831626i	\(-0.312590\pi\)
							−0.997877	+	0.0651224i	\(0.979256\pi\)
\(47\)	−1.49658	+	9.44905i	−0.218299	+	1.37829i	0.598388	+	0.801206i	\(0.295808\pi\)
							−0.816688	+	0.577080i	\(0.804192\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bs.b.46.9	yes	224
11.6	odd	10	inner	429.2.bs.b.358.9	yes	224
13.2	odd	12	inner	429.2.bs.b.145.9	yes	224
143.28	even	60	inner	429.2.bs.b.28.9	✓	224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bs.b.28.9	✓	224	143.28	even	60	inner
429.2.bs.b.46.9	yes	224	1.1	even	1	trivial
429.2.bs.b.145.9	yes	224	13.2	odd	12	inner
429.2.bs.b.358.9	yes	224	11.6	odd	10	inner