Embedded newform 456.2.bk.a.253.23

• Label: 456.2.bk.a.253.23
• 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			253.23
Character	\(\chi\)	\(=\)	456.253
Dual form			456.2.bk.a.301.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.354980 + 1.36894i) q^{2} +(-0.342020 + 0.939693i) q^{3} +(-1.74798 + 0.971890i) q^{4} +(-3.68726 + 0.650163i) q^{5} +(-1.40779 - 0.134632i) q^{6} +(2.40464 - 4.16496i) q^{7} +(-1.95095 - 2.04787i) q^{8} +(-0.766044 - 0.642788i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 6 q^{4} + 6 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.354980	+	1.36894i	0.251009	+	0.967985i
							\(3\)	−0.342020	+	0.939693i	−0.197465	+	0.542532i
\(5\)	−3.68726	+	0.650163i	−1.64899	+	0.290762i								−0.919459	−	0.393185i	\(-0.871373\pi\)
							−0.729532	+	0.683946i	\(0.760262\pi\)
\(7\)	2.40464	−	4.16496i	0.908868	−	1.57421i	0.0932283	−	0.995645i	\(-0.470281\pi\)
							0.815639	−	0.578560i	\(-0.196385\pi\)
\(11\)	0.184651	−	0.106609i	0.0556745	−	0.0321437i	−0.471904	−	0.881650i	\(-0.656433\pi\)
							0.527579	+	0.849506i	\(0.323100\pi\)
\(13\)	−0.577116	−	1.58561i	−0.160063	−	0.439770i	0.833573	−	0.552410i	\(-0.186292\pi\)
							−0.993636	+	0.112640i	\(0.964069\pi\)
\(17\)	−4.37660	+	3.67240i	−1.06148	+	0.890688i	−0.994254	−	0.107049i	\(-0.965860\pi\)
							−0.0672273	+	0.997738i	\(0.521415\pi\)
\(19\)	−4.11312	−	1.44300i	−0.943614	−	0.331047i
							\(23\)	0.281982	−	1.59920i	0.0587973	−	0.333456i	−0.941193	−	0.337869i	\(-0.890294\pi\)
0.999990	+	0.00441355i	\(0.00140488\pi\)
\(29\)	5.74365	−	6.84502i	1.06657	−	1.27109i	0.105607	−	0.994408i	\(-0.466321\pi\)
							0.960962	−	0.276680i	\(-0.0892342\pi\)
\(31\)	−2.71861	+	4.70877i	−0.488277	+	0.845721i	−0.999909	−	0.0134840i	\(-0.995708\pi\)
							0.511632	+	0.859205i	\(0.329041\pi\)
\(37\)		−	2.87551i		−	0.472730i	−0.971664	−	0.236365i	\(-0.924044\pi\)
							0.971664	−	0.236365i	\(-0.0759562\pi\)
\(41\)	−11.8130	−	4.29959i	−1.84488	−	0.671483i	−0.987676	−	0.156513i	\(-0.949975\pi\)
							−0.857208	−	0.514970i	\(-0.827803\pi\)
\(43\)	0.363742	−	0.0641374i	0.0554701	−	0.00978087i	−0.145844	−	0.989308i	\(-0.546590\pi\)
							0.201314	+	0.979527i	\(0.435479\pi\)
\(47\)	−2.38325	−	1.99978i	−0.347633	−	0.291698i	0.452206	−	0.891914i	\(-0.350637\pi\)
							−0.799839	+	0.600215i	\(0.795082\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.2.bk.a.253.23	yes	240
8.5	even	2	inner	456.2.bk.a.253.6	✓	240
19.16	even	9	inner	456.2.bk.a.301.6	yes	240
152.149	even	18	inner	456.2.bk.a.301.23	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.bk.a.253.6	✓	240	8.5	even	2	inner
456.2.bk.a.253.23	yes	240	1.1	even	1	trivial
456.2.bk.a.301.6	yes	240	19.16	even	9	inner
456.2.bk.a.301.23	yes	240	152.149	even	18	inner