Embedded newform 456.2.bk.a.253.22

• Label: 456.2.bk.a.253.22
• 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			253.22
Character	\(\chi\)	\(=\)	456.253
Dual form			456.2.bk.a.301.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0771351 - 1.41211i) q^{2} +(-0.342020 + 0.939693i) q^{3} +(-1.98810 - 0.217846i) q^{4} +(1.02705 - 0.181096i) q^{5} +(1.30057 + 0.555453i) q^{6} +(1.05714 - 1.83102i) q^{7} +(-0.460975 + 2.79061i) q^{8} +(-0.766044 - 0.642788i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 6 q^{4} + 6 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0771351	−	1.41211i	0.0545427	−	0.998511i
							\(3\)	−0.342020	+	0.939693i	−0.197465	+	0.542532i
\(5\)	1.02705	−	0.181096i	0.459309	−	0.0809886i								0.0607939	−	0.998150i	\(-0.480637\pi\)
							0.398515	+	0.917162i	\(0.369526\pi\)
\(7\)	1.05714	−	1.83102i	0.399561	−	0.692059i	−0.594111	−	0.804383i	\(-0.702496\pi\)
							0.993672	+	0.112324i	\(0.0358294\pi\)
\(11\)	−0.636064	+	0.367231i	−0.191780	+	0.110724i	−0.592816	−	0.805338i	\(-0.701984\pi\)
							0.401035	+	0.916063i	\(0.368650\pi\)
\(13\)	−1.39451	−	3.83138i	−0.386767	−	1.06263i	−0.968448	−	0.249216i	\(-0.919827\pi\)
							0.581681	−	0.813417i	\(-0.302395\pi\)
\(17\)	5.40511	−	4.53543i	1.31093	−	1.10000i	0.322788	−	0.946471i	\(-0.395380\pi\)
							0.988144	−	0.153531i	\(-0.0490644\pi\)
\(19\)	4.25650	+	0.939253i	0.976508	+	0.215480i
							\(23\)	1.20937	−	6.85866i	0.252170	−	1.43013i	−0.551063	−	0.834464i	\(-0.685777\pi\)
0.803233	−	0.595665i	\(-0.203111\pi\)
\(29\)	3.88540	−	4.63044i	0.721501	−	0.859851i	−0.273275	−	0.961936i	\(-0.588107\pi\)
							0.994776	+	0.102085i	\(0.0325513\pi\)
\(31\)	−4.59776	+	7.96355i	−0.825782	+	1.43030i	0.0755385	+	0.997143i	\(0.475932\pi\)
							−0.901320	+	0.433153i	\(0.857401\pi\)
\(37\)		−	7.26400i		−	1.19419i	−0.802169	−	0.597097i	\(-0.796321\pi\)
							0.802169	−	0.597097i	\(-0.203679\pi\)
\(41\)	−9.52355	−	3.46629i	−1.48733	−	0.541344i	−0.534584	−	0.845115i	\(-0.679532\pi\)
							−0.952745	+	0.303772i	\(0.901754\pi\)
\(43\)	−5.13395	+	0.905254i	−0.782920	+	0.138050i	−0.550800	−	0.834637i	\(-0.685677\pi\)
							−0.232120	+	0.972687i	\(0.574566\pi\)
\(47\)	4.41904	+	3.70801i	0.644583	+	0.540869i	0.905422	−	0.424513i	\(-0.139555\pi\)
							−0.260839	+	0.965382i	\(0.583999\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.2.bk.a.253.22	✓	240
8.5	even	2	inner	456.2.bk.a.253.39	yes	240
19.16	even	9	inner	456.2.bk.a.301.39	yes	240
152.149	even	18	inner	456.2.bk.a.301.22	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.bk.a.253.22	✓	240	1.1	even	1	trivial
456.2.bk.a.253.39	yes	240	8.5	even	2	inner
456.2.bk.a.301.22	yes	240	152.149	even	18	inner
456.2.bk.a.301.39	yes	240	19.16	even	9	inner