Embedded newform 229.4.e.a.135.47

• Label: 229.4.e.a.135.47
• Level: $229$
• Weight: $4$
• Character: 229.135
• Analytic conductor: $13.511$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 229 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	229.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.5114373913\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(56\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			135.47
Character	\(\chi\)	\(=\)	229.135
Dual form			229.4.e.a.95.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+3.93710i q^{2} +(4.04600 - 7.00788i) q^{3} -7.50078 q^{4} +(-0.327278 - 0.566862i) q^{5} +(27.5907 + 15.9295i) q^{6} +(8.49678 - 4.90562i) q^{7} +1.96549i q^{8} +(-19.2403 - 33.3251i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 3 q^{3} - 412 q^{4} + 12 q^{5} + 21 q^{6} - 57 q^{7} - 513 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/229\mathbb{Z}\right)^\times\).

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			3.93710i			1.39198i	0.718053	+	0.695988i	\(0.245033\pi\)
							−0.718053	+	0.695988i	\(0.754967\pi\)
\(3\)	4.04600	−	7.00788i	0.778653	−	1.34867i	−0.154065	−	0.988061i	\(-0.549236\pi\)
							0.932718	−	0.360606i	\(-0.117430\pi\)
\(5\)	−0.327278	−	0.566862i	−0.0292726	−	0.0507017i	0.851018	−	0.525137i	\(-0.175986\pi\)
							−0.880291	+	0.474435i	\(0.842652\pi\)
\(7\)	8.49678	−	4.90562i	0.458783	−	0.264879i	−0.252749	−	0.967532i	\(-0.581335\pi\)
							0.711532	+	0.702653i	\(0.248001\pi\)
\(11\)	40.8569			1.11989			0.559947	−	0.828529i	\(-0.310822\pi\)
							0.559947	+	0.828529i	\(0.310822\pi\)
\(13\)		−	53.3582i		−	1.13838i	−0.822207	−	0.569188i	\(-0.807258\pi\)
							0.822207	−	0.569188i	\(-0.192742\pi\)
\(17\)	−1.15171			−0.0164312			−0.00821559	−	0.999966i	\(-0.502615\pi\)
							−0.00821559	+	0.999966i	\(0.502615\pi\)
\(19\)	−30.0316	−	52.0162i	−0.362617	−	0.628070i	0.625774	−	0.780004i	\(-0.284783\pi\)
							−0.988391	+	0.151934i	\(0.951450\pi\)
\(23\)	48.9356	+	28.2530i	0.443642	+	0.256137i	0.705141	−	0.709067i	\(-0.250884\pi\)
							−0.261499	+	0.965204i	\(0.584217\pi\)
\(29\)	133.727	+	77.2075i	0.856295	+	0.494382i	0.862770	−	0.505597i	\(-0.168728\pi\)
							−0.00647511	+	0.999979i	\(0.502061\pi\)
\(31\)	124.035	−	71.6117i	0.718625	−	0.414898i	−0.0956213	−	0.995418i	\(-0.530484\pi\)
							0.814246	+	0.580519i	\(0.197150\pi\)
\(37\)	8.30088	−	14.3776i	0.0368826	−	0.0638826i	−0.846995	−	0.531601i	\(-0.821591\pi\)
							0.883877	+	0.467719i	\(0.154924\pi\)
\(41\)	123.724	−	71.4319i	0.471278	−	0.272092i	−0.245497	−	0.969397i	\(-0.578951\pi\)
							0.716774	+	0.697305i	\(0.245618\pi\)
\(43\)	305.296			1.08273			0.541363	−	0.840789i	\(-0.317909\pi\)
							0.541363	+	0.840789i	\(0.317909\pi\)
\(47\)	−524.748	+	302.963i	−1.62856	+	0.940250i	−0.644037	+	0.764994i	\(0.722742\pi\)
							−0.984523	+	0.175255i	\(0.943925\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	229.4.e.a.135.47	yes	112
229.95	even	6	inner	229.4.e.a.95.10	✓	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
229.4.e.a.95.10	✓	112	229.95	even	6	inner
229.4.e.a.135.47	yes	112	1.1	even	1	trivial