Embedded newform 456.2.bp.a.67.16

• Label: 456.2.bp.a.67.16
• Level: $456$
• Weight: $2$
• Character: 456.67
• Analytic conductor: $3.641$
• Analytic rank: $0$
• Dimension: $240$
• 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			67.16
Character	\(\chi\)	\(=\)	456.67
Dual form			456.2.bp.a.211.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.473247 - 1.33268i) q^{2} +(-0.642788 - 0.766044i) q^{3} +(-1.55207 + 1.26137i) q^{4} +(-0.164021 + 0.450645i) q^{5} +(-0.716695 + 1.21916i) q^{6} +(-0.0379501 - 0.0219105i) q^{7} +(2.41552 + 1.47148i) q^{8} +(-0.173648 + 0.984808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 6 q^{4} - 6 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.473247	−	1.33268i	−0.334636	−	0.942347i
							\(3\)	−0.642788	−	0.766044i	−0.371114	−	0.442276i
\(5\)	−0.164021	+	0.450645i	−0.0733526	+	0.201535i								−0.970950	−	0.239281i	\(-0.923088\pi\)
							0.897598	+	0.440815i	\(0.145311\pi\)
\(7\)	−0.0379501	−	0.0219105i	−0.0143438	−	0.00828140i	0.492811	−	0.870136i	\(-0.335969\pi\)
							−0.507155	+	0.861855i	\(0.669303\pi\)
\(11\)	2.49370	+	4.31922i	0.751879	+	1.30229i	0.946911	+	0.321496i	\(0.104186\pi\)
							−0.195032	+	0.980797i	\(0.562481\pi\)
\(13\)	0.187272	+	0.157140i	0.0519400	+	0.0435829i	0.668388	−	0.743813i	\(-0.266985\pi\)
							−0.616448	+	0.787396i	\(0.711429\pi\)
\(17\)	0.330100	+	1.87209i	0.0800610	+	0.454048i	0.998313	+	0.0580531i	\(0.0184893\pi\)
							−0.918253	+	0.395995i	\(0.870400\pi\)
\(19\)	4.35326	+	0.221578i	0.998707	+	0.0508335i
							\(23\)	−2.79913	−	7.69054i	−0.583658	−	1.60359i	−0.781879	−	0.623430i	\(-0.785738\pi\)
0.198221	−	0.980157i	\(-0.436484\pi\)
\(29\)	−1.50233	+	8.52014i	−0.278976	+	1.58215i	0.447066	+	0.894501i	\(0.352469\pi\)
							−0.726042	+	0.687650i	\(0.758642\pi\)
\(31\)	3.50499	−	6.07082i	0.629515	−	1.09035i	−0.358135	−	0.933670i	\(-0.616587\pi\)
							0.987649	−	0.156681i	\(-0.0500795\pi\)
\(37\)	10.6016			1.74290			0.871450	−	0.490485i	\(-0.163180\pi\)
							0.871450	+	0.490485i	\(0.163180\pi\)
\(41\)	5.03527	+	6.00081i	0.786378	+	0.937169i	0.999203	−	0.0399218i	\(-0.0127109\pi\)
							−0.212825	+	0.977090i	\(0.568266\pi\)
\(43\)	−2.52094	−	0.917546i	−0.384439	−	0.139924i	0.142568	−	0.989785i	\(-0.454464\pi\)
							−0.527008	+	0.849860i	\(0.676686\pi\)
\(47\)	−2.98740	−	0.526759i	−0.435757	−	0.0768357i	−0.0485335	−	0.998822i	\(-0.515455\pi\)
							−0.387224	+	0.921986i	\(0.626566\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.2.bp.a.67.16	✓	240
8.3	odd	2	inner	456.2.bp.a.67.19	yes	240
19.2	odd	18	inner	456.2.bp.a.211.19	yes	240
152.59	even	18	inner	456.2.bp.a.211.16	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.bp.a.67.16	✓	240	1.1	even	1	trivial
456.2.bp.a.67.19	yes	240	8.3	odd	2	inner
456.2.bp.a.211.16	yes	240	152.59	even	18	inner
456.2.bp.a.211.19	yes	240	19.2	odd	18	inner