Embedded newform 456.2.bk.a.301.17

• Label: 456.2.bk.a.301.17
• 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			301.17
Character	\(\chi\)	\(=\)	456.301
Dual form			456.2.bk.a.253.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.378045 - 1.36275i) q^{2} +(0.342020 + 0.939693i) q^{3} +(-1.71416 + 1.03036i) q^{4} +(-2.71912 - 0.479454i) q^{5} +(1.15127 - 0.821333i) q^{6} +(-1.39493 - 2.41609i) q^{7} +(2.05215 + 1.94645i) q^{8} +(-0.766044 + 0.642788i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 6 q^{4} + 6 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.378045	−	1.36275i	−0.267318	−	0.963608i
							\(3\)	0.342020	+	0.939693i	0.197465	+	0.542532i
\(5\)	−2.71912	−	0.479454i	−1.21603	−	0.214418i								−0.471412	−	0.881913i	\(-0.656256\pi\)
							−0.744615	+	0.667495i	\(0.767367\pi\)
\(7\)	−1.39493	−	2.41609i	−0.527234	−	0.913196i	−0.999496	−	0.0317382i	\(-0.989896\pi\)
							0.472262	−	0.881458i	\(-0.343438\pi\)
\(11\)	5.27263	+	3.04415i	1.58976	+	0.917847i	0.993346	+	0.115167i	\(0.0367404\pi\)
							0.596411	+	0.802679i	\(0.296593\pi\)
\(13\)	−0.786737	+	2.16154i	−0.218202	+	0.599504i	−0.999702	−	0.0244003i	\(-0.992232\pi\)
							0.781501	+	0.623905i	\(0.214455\pi\)
\(17\)	2.16237	+	1.81444i	0.524452	+	0.440067i	0.866180	−	0.499731i	\(-0.166568\pi\)
							−0.341729	+	0.939799i	\(0.611012\pi\)
\(19\)	−0.359711	+	4.34403i	−0.0825234	+	0.996589i
							\(23\)	1.07942	+	6.12172i	0.225075	+	1.27647i	0.862541	+	0.505987i	\(0.168872\pi\)
−0.637465	+	0.770479i	\(0.720017\pi\)
\(29\)	4.49931	+	5.36207i	0.835500	+	0.995711i	0.999957	+	0.00932617i	\(0.00296866\pi\)
							−0.164456	+	0.986384i	\(0.552587\pi\)
\(31\)	−2.36156	−	4.09034i	−0.424149	−	0.734647i	0.572192	−	0.820120i	\(-0.306093\pi\)
							−0.996341	+	0.0854726i	\(0.972760\pi\)
\(37\)			8.19803i			1.34775i	0.738846	+	0.673874i	\(0.235371\pi\)
							−0.738846	+	0.673874i	\(0.764629\pi\)
\(41\)	0.915065	−	0.333056i	0.142909	−	0.0520147i	−0.269575	−	0.962979i	\(-0.586883\pi\)
							0.412484	+	0.910965i	\(0.364661\pi\)
\(43\)	−9.76716	−	1.72221i	−1.48948	−	0.262635i	−0.631120	−	0.775685i	\(-0.717404\pi\)
							−0.858359	+	0.513050i	\(0.828515\pi\)
\(47\)	−1.90860	+	1.60150i	−0.278398	+	0.233603i	−0.771285	−	0.636490i	\(-0.780386\pi\)
							0.492888	+	0.870093i	\(0.335941\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.2.bk.a.301.17	yes	240
8.5	even	2	inner	456.2.bk.a.301.3	yes	240
19.6	even	9	inner	456.2.bk.a.253.3	✓	240
152.101	even	18	inner	456.2.bk.a.253.17	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.bk.a.253.3	✓	240	19.6	even	9	inner
456.2.bk.a.253.17	yes	240	152.101	even	18	inner
456.2.bk.a.301.3	yes	240	8.5	even	2	inner
456.2.bk.a.301.17	yes	240	1.1	even	1	trivial