Embedded newform 456.2.bk.a.253.21

• Label: 456.2.bk.a.253.21
• Level: $456$
• Weight: $2$
• Character: 456.253
• Analytic conductor: $3.641$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	456.bk (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.64117833217\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(40\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			253.21
Character	\(\chi\)	\(=\)	456.253
Dual form			456.2.bk.a.301.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.107183 + 1.41015i) q^{2} +(0.342020 - 0.939693i) q^{3} +(-1.97702 - 0.302288i) q^{4} +(3.00568 - 0.529983i) q^{5} +(1.28844 + 0.583018i) q^{6} +(1.26024 - 2.18279i) q^{7} +(0.638174 - 2.75549i) q^{8} +(-0.766044 - 0.642788i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 6 q^{4} + 6 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/456\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(229\)	\(305\)	\(343\)
\(\chi(n)\)	\(e\left(\frac{7}{9}\right)\)	\(-1\)	\(1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.107183	+	1.41015i	−0.0757900	+	0.997124i
							\(3\)	0.342020	−	0.939693i	0.197465	−	0.542532i
\(5\)	3.00568	−	0.529983i	1.34418	−	0.237016i								0.545168	−	0.838327i	\(-0.316466\pi\)
							0.799015	+	0.601311i	\(0.205355\pi\)
\(7\)	1.26024	−	2.18279i	0.476324	−	0.825018i	−0.523308	−	0.852144i	\(-0.675302\pi\)
							0.999632	+	0.0271258i	\(0.00863548\pi\)
\(11\)	−2.96253	+	1.71042i	−0.893235	+	0.515710i	−0.874999	−	0.484124i	\(-0.839138\pi\)
							−0.0182359	+	0.999834i	\(0.505805\pi\)
\(13\)	−1.65453	−	4.54580i	−0.458885	−	1.26078i	−0.926316	−	0.376747i	\(-0.877043\pi\)
							0.467431	−	0.884030i	\(-0.345180\pi\)
\(17\)	2.60141	−	2.18284i	0.630934	−	0.529416i	−0.270285	−	0.962780i	\(-0.587118\pi\)
							0.901219	+	0.433364i	\(0.142674\pi\)
\(19\)	0.378975	−	4.34239i	0.0869429	−	0.996213i
							\(23\)	−0.901451	+	5.11238i	−0.187965	+	1.06601i	0.734120	+	0.679019i	\(0.237595\pi\)
−0.922086	+	0.386986i	\(0.873516\pi\)
\(29\)	6.27764	−	7.48140i	1.16573	−	1.38926i	0.259891	−	0.965638i	\(-0.416313\pi\)
							0.905838	−	0.423624i	\(-0.139242\pi\)
\(31\)	−4.05999	+	7.03210i	−0.729195	+	1.26300i	0.228029	+	0.973654i	\(0.426772\pi\)
							−0.957224	+	0.289349i	\(0.906561\pi\)
\(37\)			11.4327i			1.87953i	0.341818	+	0.939766i	\(0.388957\pi\)
							−0.341818	+	0.939766i	\(0.611043\pi\)
\(41\)	7.17459	+	2.61134i	1.12048	+	0.407823i	0.834829	−	0.550510i	\(-0.185567\pi\)
							0.285655	+	0.958333i	\(0.407789\pi\)
\(43\)	3.23991	−	0.571283i	0.494081	−	0.0871198i	0.0789447	−	0.996879i	\(-0.474845\pi\)
							0.415136	+	0.909759i	\(0.363734\pi\)
\(47\)	7.30468	+	6.12936i	1.06550	+	0.894059i	0.994637	−	0.103428i	\(-0.0329810\pi\)
							0.0708605	+	0.997486i	\(0.477425\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.2.bk.a.253.21	yes	240
8.5	even	2	inner	456.2.bk.a.253.4	✓	240
19.16	even	9	inner	456.2.bk.a.301.4	yes	240
152.149	even	18	inner	456.2.bk.a.301.21	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.bk.a.253.4	✓	240	8.5	even	2	inner
456.2.bk.a.253.21	yes	240	1.1	even	1	trivial
456.2.bk.a.301.4	yes	240	19.16	even	9	inner
456.2.bk.a.301.21	yes	240	152.149	even	18	inner