Embedded newform 43.8.g.a.9.10

• Label: 43.8.g.a.9.10
• 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			9.10
Character	\(\chi\)	\(=\)	43.9
Dual form			43.8.g.a.24.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.70223 - 3.38849i) q^{2} +(-50.7403 - 7.64788i) q^{3} +(24.3029 - 106.478i) q^{4} +(-388.722 - 265.026i) q^{5} +(111.197 + 192.599i) q^{6} +(811.865 - 1406.19i) q^{7} +(-926.290 + 446.078i) q^{8} +(426.254 + 131.482i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 40 q^{2} - 12 q^{3} - 3028 q^{4} + 235 q^{5} + 1274 q^{6} - 4122 q^{7} + 762 q^{8} + 8097 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.70223	−	3.38849i	−0.238846	−	0.299503i	0.647933	−	0.761697i	\(-0.275634\pi\)
							−0.886779	+	0.462194i	\(0.847062\pi\)
\(3\)	−50.7403	−	7.64788i	−1.08500	−	0.163537i	−0.417890	−	0.908498i	\(-0.637230\pi\)
							−0.667108	+	0.744961i	\(0.732468\pi\)
\(5\)	−388.722	−	265.026i	−1.39073	−	0.948187i	−0.999612	−	0.0278477i	\(-0.991135\pi\)
							−0.391122	−	0.920339i	\(-0.627913\pi\)
\(7\)	811.865	−	1406.19i	0.894624	−	1.54953i	0.0603557	−	0.998177i	\(-0.480777\pi\)
							0.834269	−	0.551358i	\(-0.185890\pi\)
\(11\)	−272.163	−	1192.42i	−0.0616530	−	0.270120i	0.934701	−	0.355435i	\(-0.115667\pi\)
							−0.996354	+	0.0853156i	\(0.972810\pi\)
\(13\)	377.290	−	5034.58i	0.0476292	−	0.635567i	−0.921399	−	0.388618i	\(-0.872953\pi\)
							0.969028	−	0.246950i	\(-0.0794282\pi\)
\(17\)	−14540.7	+	9913.68i	−0.717817	+	0.489400i	−0.866305	−	0.499515i	\(-0.833512\pi\)
							0.148488	+	0.988914i	\(0.452559\pi\)
\(19\)	25096.2	−	7741.15i	0.839402	−	0.258921i	0.154898	−	0.987930i	\(-0.450495\pi\)
							0.684504	+	0.729009i	\(0.260019\pi\)
\(23\)	6932.73	−	6432.64i	0.118811	−	0.110241i	−0.618530	−	0.785761i	\(-0.712272\pi\)
							0.737341	+	0.675521i	\(0.236081\pi\)
\(29\)	50903.1	−	7672.40i	0.387571	−	0.0584169i	0.0476331	−	0.998865i	\(-0.484832\pi\)
							0.339938	+	0.940448i	\(0.389594\pi\)
\(31\)	89334.4	−	227620.i	0.538583	−	1.37229i	−0.359782	−	0.933037i	\(-0.617149\pi\)
							0.898365	−	0.439250i	\(-0.144756\pi\)
\(37\)	−45569.4	−	78928.5i	−0.147900	−	0.256170i	0.782551	−	0.622586i	\(-0.213918\pi\)
							−0.930451	+	0.366416i	\(0.880585\pi\)
\(41\)	−153062.	−	191934.i	−0.346836	−	0.434919i	0.577562	−	0.816347i	\(-0.304004\pi\)
							−0.924399	+	0.381427i	\(0.875433\pi\)
\(43\)	489022.	−	180764.i	0.937971	−	0.346715i
							\(47\)	277064.	−	1.21390e6i	0.389258	−	1.70545i	−0.277964	−	0.960592i	\(-0.589659\pi\)
0.667222	−	0.744859i	\(-0.267483\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.g.a.9.10	✓	300
43.24	even	21	inner	43.8.g.a.24.10	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.g.a.9.10	✓	300	1.1	even	1	trivial
43.8.g.a.24.10	yes	300	43.24	even	21	inner