Embedded newform 43.8.g.a.10.17

• Label: 43.8.g.a.10.17
• Level: $43$
• Weight: $8$
• Character: 43.10
• Analytic conductor: $13.433$
• Analytic rank: $0$
• Dimension: $300$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.4325560958\)
Analytic rank:	\(0\)
Dimension:	\(300\)
Relative dimension:	\(25\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			10.17
Character	\(\chi\)	\(=\)	43.10
Dual form			43.8.g.a.13.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.77056 + 7.75731i) q^{2} +(28.3063 + 26.2644i) q^{3} +(58.2830 - 28.0676i) q^{4} +(5.92871 - 0.893610i) q^{5} +(-153.623 + 266.083i) q^{6} +(270.752 + 468.955i) q^{7} +(955.929 + 1198.70i) q^{8} +(-52.0073 - 693.989i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 40 q^{2} - 12 q^{3} - 3028 q^{4} + 235 q^{5} + 1274 q^{6} - 4122 q^{7} + 762 q^{8} + 8097 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.77056	+	7.75731i	0.156497	+	0.685656i	0.990911	+	0.134519i	\(0.0429488\pi\)
							−0.834415	+	0.551137i	\(0.814194\pi\)
\(3\)	28.3063	+	26.2644i	0.605283	+	0.561621i	0.922142	−	0.386852i	\(-0.126438\pi\)
							−0.316859	+	0.948473i	\(0.602628\pi\)
\(5\)	5.92871	−	0.893610i	0.0212112	−	0.00319708i	−0.138428	−	0.990372i	\(-0.544205\pi\)
							0.159639	+	0.987175i	\(0.448967\pi\)
\(7\)	270.752	+	468.955i	0.298351	+	0.516759i	0.975759	−	0.218848i	\(-0.0702300\pi\)
							−0.677408	+	0.735608i	\(0.736897\pi\)
\(11\)	4933.83	+	2376.01i	1.11766	+	0.538237i	0.899169	−	0.437602i	\(-0.144172\pi\)
							0.218492	+	0.975839i	\(0.429886\pi\)
\(13\)	581.542	−	1481.74i	0.0734140	−	0.187056i	−0.889456	−	0.457020i	\(-0.848917\pi\)
							0.962870	+	0.269964i	\(0.0870119\pi\)
\(17\)	−16844.0	−	2538.83i	−0.831524	−	0.125332i	−0.280534	−	0.959844i	\(-0.590512\pi\)
							−0.550990	+	0.834512i	\(0.685750\pi\)
\(19\)	−3560.49	+	47511.5i	−0.119089	+	1.58913i	0.543015	+	0.839723i	\(0.317283\pi\)
							−0.662104	+	0.749412i	\(0.730336\pi\)
\(23\)	37229.0	−	25382.3i	0.638020	−	0.434994i	−0.200629	−	0.979667i	\(-0.564299\pi\)
							0.838649	+	0.544673i	\(0.183346\pi\)
\(29\)	−125219.	+	116186.i	−0.953404	+	0.884629i	−0.993415	−	0.114570i	\(-0.963451\pi\)
							0.0400115	+	0.999199i	\(0.487261\pi\)
\(31\)	−44702.1	−	13788.8i	−0.269502	−	0.0831304i	0.157059	−	0.987589i	\(-0.449799\pi\)
							−0.426561	+	0.904459i	\(0.640275\pi\)
\(37\)	150256.	−	260251.i	0.487671	−	0.844670i	−0.512229	−	0.858849i	\(-0.671180\pi\)
							0.999899	+	0.0141788i	\(0.00451341\pi\)
\(41\)	−138300.	−	605932.i	−0.313386	−	1.37303i	−0.848921	−	0.528519i	\(-0.822747\pi\)
							0.535536	−	0.844512i	\(-0.320110\pi\)
\(43\)	518359.	−	55879.9i	0.994240	−	0.107181i
							\(47\)	−85504.9	+	41177.0i	−0.120129	+	0.0578512i	−0.492983	−	0.870039i	\(-0.664094\pi\)
0.372854	+	0.927890i	\(0.378379\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.g.a.10.17	✓	300
43.13	even	21	inner	43.8.g.a.13.17	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.g.a.10.17	✓	300	1.1	even	1	trivial
43.8.g.a.13.17	yes	300	43.13	even	21	inner