Embedded newform 456.2.bk.a.301.37

• Label: 456.2.bk.a.301.37
• Level: $456$
• Weight: $2$
• Character: 456.301
• 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			301.37
Character	\(\chi\)	\(=\)	456.301
Dual form			456.2.bk.a.253.37

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36002 - 0.387744i) q^{2} +(-0.342020 - 0.939693i) q^{3} +(1.69931 - 1.05468i) q^{4} +(2.74544 + 0.484095i) q^{5} +(-0.829515 - 1.14538i) q^{6} +(-1.04707 - 1.81357i) q^{7} +(1.90215 - 2.09328i) q^{8} +(-0.766044 + 0.642788i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 6 q^{4} + 6 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(229\)	\(305\)	\(343\)
\(\chi(n)\)	\(e\left(\frac{2}{9}\right)\)	\(-1\)	\(1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.36002	−	0.387744i	0.961679	−	0.274177i
							\(3\)	−0.342020	−	0.939693i	−0.197465	−	0.542532i
\(5\)	2.74544	+	0.484095i	1.22780	+	0.216494i								0.749679	−	0.661802i	\(-0.230208\pi\)
							0.478118	+	0.878296i	\(0.341319\pi\)
\(7\)	−1.04707	−	1.81357i	−0.395754	−	0.685466i	0.597443	−	0.801911i	\(-0.296183\pi\)
							−0.993197	+	0.116445i	\(0.962850\pi\)
\(11\)	−0.893712	−	0.515985i	−0.269464	−	0.155575i	0.359180	−	0.933268i	\(-0.383056\pi\)
							−0.628644	+	0.777693i	\(0.716390\pi\)
\(13\)	−1.06002	+	2.91237i	−0.293996	+	0.807746i	0.701476	+	0.712693i	\(0.252525\pi\)
							−0.995472	+	0.0950538i	\(0.969698\pi\)
\(17\)	0.735497	+	0.617155i	0.178384	+	0.149682i	0.727608	−	0.685993i	\(-0.240632\pi\)
							−0.549224	+	0.835675i	\(0.685077\pi\)
\(19\)	−0.854759	+	4.27427i	−0.196095	+	0.980585i
							\(23\)	−0.402778	−	2.28427i	−0.0839851	−	0.476303i	−0.997571	−	0.0696573i	\(-0.977809\pi\)
0.913586	−	0.406646i	\(-0.133302\pi\)
\(29\)	2.18766	+	2.60715i	0.406238	+	0.484136i	0.929911	−	0.367784i	\(-0.119883\pi\)
							−0.523673	+	0.851919i	\(0.675439\pi\)
\(31\)	−2.85218	−	4.94012i	−0.512267	−	0.887273i	−0.999899	−	0.0142231i	\(-0.995472\pi\)
							0.487632	−	0.873049i	\(-0.337861\pi\)
\(37\)			3.92447i			0.645178i	0.946539	+	0.322589i	\(0.104553\pi\)
							−0.946539	+	0.322589i	\(0.895447\pi\)
\(41\)	−3.19432	+	1.16264i	−0.498870	+	0.181574i	−0.579185	−	0.815196i	\(-0.696629\pi\)
							0.0803159	+	0.996769i	\(0.474407\pi\)
\(43\)	8.90791	+	1.57071i	1.35844	+	0.239530i	0.804960	−	0.593329i	\(-0.202187\pi\)
							0.553484	+	0.832860i	\(0.313298\pi\)
\(47\)	−7.39087	+	6.20168i	−1.07807	+	0.904608i	−0.995759	−	0.0919987i	\(-0.970674\pi\)
							−0.0823108	+	0.996607i	\(0.526230\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.2.bk.a.301.37	yes	240
8.5	even	2	inner	456.2.bk.a.301.19	yes	240
19.6	even	9	inner	456.2.bk.a.253.19	✓	240
152.101	even	18	inner	456.2.bk.a.253.37	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.bk.a.253.19	✓	240	19.6	even	9	inner
456.2.bk.a.253.37	yes	240	152.101	even	18	inner
456.2.bk.a.301.19	yes	240	8.5	even	2	inner
456.2.bk.a.301.37	yes	240	1.1	even	1	trivial