Embedded newform 228.3.b.e.227.2

• Label: 228.3.b.e.227.2
• Level: $228$
• Weight: $3$
• Character: 228.227
• Analytic conductor: $6.213$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	228.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.21255002741\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			227.2
Character	\(\chi\)	\(=\)	228.227
Dual form			228.3.b.e.227.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99370 - 0.158675i) q^{2} +(1.15371 + 2.76929i) q^{3} +(3.94964 + 0.632700i) q^{4} +8.12041i q^{5} +(-1.86073 - 5.70418i) q^{6} +8.99393i q^{7} +(-7.77400 - 1.88812i) q^{8} +(-6.33790 + 6.38992i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 16 q^{4} + 6 q^{6} - 48 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/228\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(97\)	\(115\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99370	−	0.158675i	−0.996848	−	0.0793375i
							\(3\)	1.15371	+	2.76929i	0.384571	+	0.923096i
\(5\)			8.12041i			1.62408i								0.583600	+	0.812041i	\(0.301643\pi\)
							−0.583600	+	0.812041i	\(0.698357\pi\)
\(7\)			8.99393i			1.28485i	0.766350	+	0.642424i	\(0.222071\pi\)
							−0.766350	+	0.642424i	\(0.777929\pi\)
\(11\)	0.680370			0.0618519			0.0309259	−	0.999522i	\(-0.490154\pi\)
							0.0309259	+	0.999522i	\(0.490154\pi\)
\(13\)		−	4.64238i		−	0.357106i	−0.983930	−	0.178553i	\(-0.942858\pi\)
							0.983930	−	0.178553i	\(-0.0571416\pi\)
\(17\)		−	21.2842i		−	1.25201i	−0.779819	−	0.626005i	\(-0.784689\pi\)
							0.779819	−	0.626005i	\(-0.215311\pi\)
\(19\)	10.8345	−	15.6081i	0.570235	−	0.821481i
							\(23\)	34.5295			1.50128			0.750642	−	0.660709i	\(-0.229744\pi\)
0.750642	+	0.660709i	\(0.229744\pi\)
\(29\)	−28.9611			−0.998659			−0.499329	−	0.866412i	\(-0.666420\pi\)
							−0.499329	+	0.866412i	\(0.666420\pi\)
\(31\)	38.0395			1.22708			0.613540	−	0.789664i	\(-0.289745\pi\)
							0.613540	+	0.789664i	\(0.289745\pi\)
\(37\)			50.9685i			1.37753i	0.724986	+	0.688764i	\(0.241846\pi\)
							−0.724986	+	0.688764i	\(0.758154\pi\)
\(41\)	−41.0318			−1.00078			−0.500388	−	0.865801i	\(-0.666809\pi\)
							−0.500388	+	0.865801i	\(0.666809\pi\)
\(43\)			47.4246i			1.10290i	0.834209	+	0.551449i	\(0.185925\pi\)
							−0.834209	+	0.551449i	\(0.814075\pi\)
\(47\)	15.0004			0.319157			0.159578	−	0.987185i	\(-0.448987\pi\)
							0.159578	+	0.987185i	\(0.448987\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	228.3.b.e.227.2	yes	72
3.2	odd	2	inner	228.3.b.e.227.72	yes	72
4.3	odd	2	inner	228.3.b.e.227.3	yes	72
12.11	even	2	inner	228.3.b.e.227.69	yes	72
19.18	odd	2	inner	228.3.b.e.227.71	yes	72
57.56	even	2	inner	228.3.b.e.227.1	✓	72
76.75	even	2	inner	228.3.b.e.227.70	yes	72
228.227	odd	2	inner	228.3.b.e.227.4	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.3.b.e.227.1	✓	72	57.56	even	2	inner
228.3.b.e.227.2	yes	72	1.1	even	1	trivial
228.3.b.e.227.3	yes	72	4.3	odd	2	inner
228.3.b.e.227.4	yes	72	228.227	odd	2	inner
228.3.b.e.227.69	yes	72	12.11	even	2	inner
228.3.b.e.227.70	yes	72	76.75	even	2	inner
228.3.b.e.227.71	yes	72	19.18	odd	2	inner
228.3.b.e.227.72	yes	72	3.2	odd	2	inner