Embedded newform 43.8.g.a.9.8

• Label: 43.8.g.a.9.8
• Level: $43$
• Weight: $8$
• Character: 43.9
• 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			9.8
Character	\(\chi\)	\(=\)	43.9
Dual form			43.8.g.a.24.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.35832 - 7.97308i) q^{2} +(-51.9023 - 7.82301i) q^{3} +(5.34089 - 23.4000i) q^{4} +(-119.077 - 81.1857i) q^{5} +(267.638 + 463.562i) q^{6} +(-639.426 + 1107.52i) q^{7} +(-1396.60 + 672.566i) q^{8} +(542.807 + 167.434i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 40 q^{2} - 12 q^{3} - 3028 q^{4} + 235 q^{5} + 1274 q^{6} - 4122 q^{7} + 762 q^{8} + 8097 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.35832	−	7.97308i	−0.562001	−	0.704727i	0.416925	−	0.908941i	\(-0.363108\pi\)
							−0.978926	+	0.204213i	\(0.934536\pi\)
\(3\)	−51.9023	−	7.82301i	−1.10984	−	0.167282i	−0.431561	−	0.902084i	\(-0.642037\pi\)
							−0.678282	+	0.734801i	\(0.737275\pi\)
\(5\)	−119.077	−	81.1857i	−0.426025	−	0.290459i	0.331265	−	0.943538i	\(-0.392525\pi\)
							−0.757290	+	0.653079i	\(0.773477\pi\)
\(7\)	−639.426	+	1107.52i	−0.704608	+	1.22042i	0.262225	+	0.965007i	\(0.415544\pi\)
							−0.966833	+	0.255410i	\(0.917790\pi\)
\(11\)	−1897.71	−	8314.43i	−0.429889	−	1.88347i	−0.467179	−	0.884163i	\(-0.654730\pi\)
							0.0372903	−	0.999304i	\(-0.488127\pi\)
\(13\)	−777.239	+	10371.5i	−0.0981189	+	1.30931i	0.703167	+	0.711024i	\(0.251768\pi\)
							−0.801286	+	0.598281i	\(0.795851\pi\)
\(17\)	13557.1	−	9243.06i	0.669260	−	0.456294i	−0.180422	−	0.983589i	\(-0.557746\pi\)
							0.849682	+	0.527296i	\(0.176794\pi\)
\(19\)	15757.4	−	4860.50i	0.527043	−	0.162571i	−0.0198039	−	0.999804i	\(-0.506304\pi\)
							0.546847	+	0.837233i	\(0.315828\pi\)
\(23\)	34949.4	−	32428.3i	0.598952	−	0.555747i	−0.321347	−	0.946962i	\(-0.604136\pi\)
							0.920299	+	0.391215i	\(0.127945\pi\)
\(29\)	−197489.	+	29766.6i	−1.50366	+	0.226640i	−0.848639	−	0.528972i	\(-0.822578\pi\)
							−0.655019	+	0.755612i	\(0.727339\pi\)
\(31\)	25635.1	−	65317.2i	0.154550	−	0.393787i	−0.832537	−	0.553969i	\(-0.813113\pi\)
							0.987088	+	0.160182i	\(0.0512080\pi\)
\(37\)	−11693.9	−	20254.5i	−0.0379538	−	0.0657378i	0.846425	−	0.532509i	\(-0.178751\pi\)
							−0.884378	+	0.466771i	\(0.845417\pi\)
\(41\)	343566.	+	430818.i	0.778514	+	0.976226i	0.999999	+	0.00106196i	\(0.000338032\pi\)
							−0.221485	+	0.975164i	\(0.571091\pi\)
\(43\)	−31124.4	+	520432.i	−0.0596983	+	0.998216i
							\(47\)	−243658.	+	1.06754e6i	−0.342325	+	1.49982i	0.451828	+	0.892105i	\(0.350772\pi\)
−0.794153	+	0.607718i	\(0.792085\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.g.a.9.8	✓	300
43.24	even	21	inner	43.8.g.a.24.8	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.g.a.9.8	✓	300	1.1	even	1	trivial
43.8.g.a.24.8	yes	300	43.24	even	21	inner