Embedded newform 456.2.bk.a.253.13

• Label: 456.2.bk.a.253.13
• 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			253.13
Character	\(\chi\)	\(=\)	456.253
Dual form			456.2.bk.a.301.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.873464 - 1.11223i) q^{2} +(-0.342020 + 0.939693i) q^{3} +(-0.474121 + 1.94299i) q^{4} +(-3.30436 + 0.582649i) q^{5} +(1.34390 - 0.440382i) q^{6} +(-1.96330 + 3.40053i) q^{7} +(2.57518 - 1.16980i) q^{8} +(-0.766044 - 0.642788i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 6 q^{4} + 6 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.873464	−	1.11223i	−0.617632	−	0.786467i
							\(3\)	−0.342020	+	0.939693i	−0.197465	+	0.542532i
\(5\)	−3.30436	+	0.582649i	−1.47776	+	0.260568i								−0.853681	−	0.520796i	\(-0.825635\pi\)
							−0.624075	+	0.781364i	\(0.714524\pi\)
\(7\)	−1.96330	+	3.40053i	−0.742057	+	1.28528i	0.209501	+	0.977809i	\(0.432816\pi\)
							−0.951557	+	0.307471i	\(0.900517\pi\)
\(11\)	2.32580	−	1.34280i	0.701254	−	0.404869i	−0.106560	−	0.994306i	\(-0.533984\pi\)
							0.807814	+	0.589437i	\(0.200650\pi\)
\(13\)	−1.43956	−	3.95517i	−0.399263	−	1.09697i	−0.962644	−	0.270769i	\(-0.912722\pi\)
							0.563381	−	0.826197i	\(-0.309500\pi\)
\(17\)	4.40701	−	3.69792i	1.06886	−	0.896878i	0.0739095	−	0.997265i	\(-0.476452\pi\)
							0.994949	+	0.100387i	\(0.0320079\pi\)
\(19\)	−3.87216	+	2.00160i	−0.888334	+	0.459198i
							\(23\)	0.958194	−	5.43419i	0.199797	−	1.13311i	−0.705622	−	0.708588i	\(-0.749332\pi\)
0.905420	−	0.424518i	\(-0.139556\pi\)
\(29\)	−0.439779	+	0.524109i	−0.0816650	+	0.0973245i	−0.805330	−	0.592827i	\(-0.798012\pi\)
							0.723665	+	0.690152i	\(0.242456\pi\)
\(31\)	1.25037	−	2.16570i	0.224573	−	0.388971i	−0.731619	−	0.681714i	\(-0.761235\pi\)
							0.956191	+	0.292743i	\(0.0945681\pi\)
\(37\)			4.16521i			0.684757i	0.939562	+	0.342379i	\(0.111233\pi\)
							−0.939562	+	0.342379i	\(0.888767\pi\)
\(41\)	−10.3777	−	3.77718i	−1.62073	−	0.589897i	−0.637208	−	0.770692i	\(-0.719911\pi\)
							−0.983521	+	0.180795i	\(0.942133\pi\)
\(43\)	12.0055	−	2.11689i	1.83082	−	0.322823i	0.851378	−	0.524553i	\(-0.175767\pi\)
							0.979441	+	0.201730i	\(0.0646563\pi\)
\(47\)	4.55811	+	3.82471i	0.664869	+	0.557891i	0.911542	−	0.411207i	\(-0.134893\pi\)
							−0.246673	+	0.969099i	\(0.579337\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.2.bk.a.253.13	✓	240
8.5	even	2	inner	456.2.bk.a.253.30	yes	240
19.16	even	9	inner	456.2.bk.a.301.30	yes	240
152.149	even	18	inner	456.2.bk.a.301.13	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.bk.a.253.13	✓	240	1.1	even	1	trivial
456.2.bk.a.253.30	yes	240	8.5	even	2	inner
456.2.bk.a.301.13	yes	240	152.149	even	18	inner
456.2.bk.a.301.30	yes	240	19.16	even	9	inner