Embedded newform 456.2.bk.a.253.5

• Label: 456.2.bk.a.253.5
• 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			253.5
Character	\(\chi\)	\(=\)	456.253
Dual form			456.2.bk.a.301.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39786 + 0.214444i) q^{2} +(-0.342020 + 0.939693i) q^{3} +(1.90803 - 0.599526i) q^{4} +(2.07565 - 0.365992i) q^{5} +(0.276585 - 1.38690i) q^{6} +(0.994443 - 1.72243i) q^{7} +(-2.53859 + 1.24722i) q^{8} +(-0.766044 - 0.642788i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 6 q^{4} + 6 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39786	+	0.214444i	−0.988437	+	0.151635i
							\(3\)	−0.342020	+	0.939693i	−0.197465	+	0.542532i
\(5\)	2.07565	−	0.365992i	0.928257	−	0.163677i								0.310975	−	0.950418i	\(-0.399345\pi\)
							0.617283	+	0.786741i	\(0.288234\pi\)
\(7\)	0.994443	−	1.72243i	0.375864	−	0.651016i	−0.614592	−	0.788845i	\(-0.710679\pi\)
							0.990456	+	0.137830i	\(0.0440126\pi\)
\(11\)	4.64743	−	2.68320i	1.40125	−	0.809015i	0.406733	−	0.913547i	\(-0.366668\pi\)
							0.994521	+	0.104533i	\(0.0333347\pi\)
\(13\)	−2.31010	−	6.34696i	−0.640707	−	1.76033i	−0.649493	−	0.760367i	\(-0.725019\pi\)
							0.00878576	−	0.999961i	\(-0.497203\pi\)
\(17\)	−4.80442	+	4.03138i	−1.16524	+	0.977754i	−0.999964	−	0.00849248i	\(-0.997297\pi\)
							−0.165278	+	0.986247i	\(0.552852\pi\)
\(19\)	−1.64951	−	4.03474i	−0.378424	−	0.925632i
							\(23\)	0.307565	−	1.74429i	0.0641318	−	0.363710i	−0.935806	−	0.352517i	\(-0.885326\pi\)
0.999937	−	0.0111929i	\(-0.00356289\pi\)
\(29\)	−1.05844	+	1.26140i	−0.196548	+	0.234237i	−0.855312	−	0.518112i	\(-0.826635\pi\)
							0.658765	+	0.752349i	\(0.271079\pi\)
\(31\)	1.98429	−	3.43688i	0.356388	−	0.617282i	−0.630966	−	0.775810i	\(-0.717341\pi\)
							0.987355	+	0.158528i	\(0.0506748\pi\)
\(37\)			0.502359i			0.0825874i	0.999147	+	0.0412937i	\(0.0131479\pi\)
							−0.999147	+	0.0412937i	\(0.986852\pi\)
\(41\)	0.821603	+	0.299039i	0.128313	+	0.0467020i	0.405378	−	0.914149i	\(-0.367140\pi\)
							−0.277066	+	0.960851i	\(0.589362\pi\)
\(43\)	11.4809	−	2.02439i	1.75082	−	0.308716i	0.795863	−	0.605476i	\(-0.207017\pi\)
							0.954952	+	0.296760i	\(0.0959063\pi\)
\(47\)	4.46708	+	3.74833i	0.651591	+	0.546749i	0.907553	−	0.419937i	\(-0.137948\pi\)
							−0.255963	+	0.966687i	\(0.582392\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.2.bk.a.253.5	✓	240
8.5	even	2	inner	456.2.bk.a.253.16	yes	240
19.16	even	9	inner	456.2.bk.a.301.16	yes	240
152.149	even	18	inner	456.2.bk.a.301.5	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.bk.a.253.5	✓	240	1.1	even	1	trivial
456.2.bk.a.253.16	yes	240	8.5	even	2	inner
456.2.bk.a.301.5	yes	240	152.149	even	18	inner
456.2.bk.a.301.16	yes	240	19.16	even	9	inner