Embedded newform 229.4.e.a.95.10

• Label: 229.4.e.a.95.10
• Level: $229$
• Weight: $4$
• Character: 229.95
• 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			95.10
Character	\(\chi\)	\(=\)	229.95
Dual form			229.4.e.a.135.47

$q$-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 \(\left(\mathbb{Z}/229\mathbb{Z}\right)^\times\).

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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	\(-0.754967\pi\)
							0.718053	−	0.695988i	\(-0.245033\pi\)
\(3\)	4.04600	+	7.00788i	0.778653	+	1.34867i	0.932718	+	0.360606i	\(0.117430\pi\)
							−0.154065	+	0.988061i	\(0.549236\pi\)
\(5\)	−0.327278	+	0.566862i	−0.0292726	+	0.0507017i	−0.880291	−	0.474435i	\(-0.842652\pi\)
							0.851018	+	0.525137i	\(0.175986\pi\)
\(7\)	8.49678	+	4.90562i	0.458783	+	0.264879i	0.711532	−	0.702653i	\(-0.248001\pi\)
							−0.252749	+	0.967532i	\(0.581335\pi\)
\(11\)	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	\(0.192742\pi\)
							−0.822207	+	0.569188i	\(0.807258\pi\)
\(17\)	−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	−0.988391	−	0.151934i	\(-0.951450\pi\)
							0.625774	+	0.780004i	\(0.284783\pi\)
\(23\)	48.9356	−	28.2530i	0.443642	−	0.256137i	−0.261499	−	0.965204i	\(-0.584217\pi\)
							0.705141	+	0.709067i	\(0.250884\pi\)
\(29\)	133.727	−	77.2075i	0.856295	−	0.494382i	−0.00647511	−	0.999979i	\(-0.502061\pi\)
							0.862770	+	0.505597i	\(0.168728\pi\)
\(31\)	124.035	+	71.6117i	0.718625	+	0.414898i	0.814246	−	0.580519i	\(-0.197150\pi\)
							−0.0956213	+	0.995418i	\(0.530484\pi\)
\(37\)	8.30088	+	14.3776i	0.0368826	+	0.0638826i	0.883877	−	0.467719i	\(-0.154924\pi\)
							−0.846995	+	0.531601i	\(0.821591\pi\)
\(41\)	123.724	+	71.4319i	0.471278	+	0.272092i	0.716774	−	0.697305i	\(-0.245618\pi\)
							−0.245497	+	0.969397i	\(0.578951\pi\)
\(43\)	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	−0.984523	−	0.175255i	\(-0.943925\pi\)
							−0.644037	−	0.764994i	\(-0.722742\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
229.4.e.a.95.10	✓	112	1.1	even	1	trivial
229.4.e.a.135.47	yes	112	229.135	even	6	inner