Embedded newform 456.2.bp.a.67.14

• Label: 456.2.bp.a.67.14
• Level: $456$
• Weight: $2$
• Character: 456.67
• 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.bp (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			67.14
Character	\(\chi\)	\(=\)	456.67
Dual form			456.2.bp.a.211.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.767022 - 1.18814i) q^{2} +(0.642788 + 0.766044i) q^{3} +(-0.823355 + 1.82266i) q^{4} +(-0.425460 + 1.16894i) q^{5} +(0.417136 - 1.35129i) q^{6} +(-0.404754 - 0.233685i) q^{7} +(2.79711 - 0.419758i) q^{8} +(-0.173648 + 0.984808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 6 q^{4} - 6 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(229\)	\(305\)	\(343\)
\(\chi(n)\)	\(e\left(\frac{17}{18}\right)\)	\(-1\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.767022	−	1.18814i	−0.542366	−	0.840142i
							\(3\)	0.642788	+	0.766044i	0.371114	+	0.442276i
\(5\)	−0.425460	+	1.16894i	−0.190272	+	0.522767i								−0.997744	−	0.0671388i	\(-0.978613\pi\)
							0.807472	+	0.589906i	\(0.200835\pi\)
\(7\)	−0.404754	−	0.233685i	−0.152983	−	0.0883245i	0.421555	−	0.906803i	\(-0.361485\pi\)
							−0.574537	+	0.818479i	\(0.694818\pi\)
\(11\)	0.995900	+	1.72495i	0.300275	+	0.520092i	0.976198	−	0.216881i	\(-0.0695882\pi\)
							−0.675923	+	0.736972i	\(0.736255\pi\)
\(13\)	−3.03454	−	2.54628i	−0.841630	−	0.706211i	0.116300	−	0.993214i	\(-0.462897\pi\)
							−0.957930	+	0.287003i	\(0.907341\pi\)
\(17\)	1.03784	+	5.88589i	0.251714	+	1.42754i	0.804370	+	0.594129i	\(0.202503\pi\)
							−0.552656	+	0.833409i	\(0.686386\pi\)
\(19\)	−2.51994	+	3.55667i	−0.578115	+	0.815955i
							\(23\)	1.37286	+	3.77191i	0.286262	+	0.786497i	0.996581	+	0.0826179i	\(0.0263281\pi\)
−0.710320	+	0.703879i	\(0.751450\pi\)
\(29\)	−1.00076	+	5.67557i	−0.185836	+	1.05393i	0.739041	+	0.673660i	\(0.235279\pi\)
							−0.924877	+	0.380267i	\(0.875832\pi\)
\(31\)	0.534453	−	0.925699i	0.0959905	−	0.166260i	−0.814031	−	0.580821i	\(-0.802731\pi\)
							0.910022	+	0.414561i	\(0.136065\pi\)
\(37\)	−10.4930			−1.72503			−0.862516	−	0.506029i	\(-0.831113\pi\)
							−0.862516	+	0.506029i	\(0.831113\pi\)
\(41\)	−2.73445	−	3.25879i	−0.427049	−	0.508938i	0.509019	−	0.860755i	\(-0.330008\pi\)
							−0.936069	+	0.351818i	\(0.885564\pi\)
\(43\)	6.67888	+	2.43091i	1.01852	+	0.370711i	0.796699	−	0.604377i	\(-0.206578\pi\)
							0.221821	+	0.975088i	\(0.428800\pi\)
\(47\)	8.83316	+	1.55752i	1.28845	+	0.227188i	0.775564	−	0.631269i	\(-0.217466\pi\)
							0.512885	+	0.858457i	\(0.328577\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.2.bp.a.67.14	✓	240
8.3	odd	2	inner	456.2.bp.a.67.18	yes	240
19.2	odd	18	inner	456.2.bp.a.211.18	yes	240
152.59	even	18	inner	456.2.bp.a.211.14	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.bp.a.67.14	✓	240	1.1	even	1	trivial
456.2.bp.a.67.18	yes	240	8.3	odd	2	inner
456.2.bp.a.211.14	yes	240	152.59	even	18	inner
456.2.bp.a.211.18	yes	240	19.2	odd	18	inner