Embedded newform 43.8.g.a.10.22

• Label: 43.8.g.a.10.22
• 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			10.22
Character	\(\chi\)	\(=\)	43.10
Dual form			43.8.g.a.13.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.96114 + 17.3549i) q^{2} +(-6.73699 - 6.25101i) q^{3} +(-170.178 + 81.9534i) q^{4} +(429.420 - 64.7247i) q^{5} +(81.7996 - 141.681i) q^{6} +(837.540 + 1450.66i) q^{7} +(-675.736 - 847.347i) q^{8} +(-157.123 - 2096.66i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 40 q^{2} - 12 q^{3} - 3028 q^{4} + 235 q^{5} + 1274 q^{6} - 4122 q^{7} + 762 q^{8} + 8097 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.96114	+	17.3549i	0.350119	+	1.53397i	0.776905	+	0.629618i	\(0.216789\pi\)
							−0.426786	+	0.904353i	\(0.640354\pi\)
\(3\)	−6.73699	−	6.25101i	−0.144059	−	0.133668i	0.604852	−	0.796338i	\(-0.293232\pi\)
							−0.748911	+	0.662671i	\(0.769423\pi\)
\(5\)	429.420	−	64.7247i	1.53634	−	0.231566i	0.674320	−	0.738440i	\(-0.264437\pi\)
							0.862020	+	0.506874i	\(0.169199\pi\)
\(7\)	837.540	+	1450.66i	0.922916	+	1.59854i	0.794879	+	0.606768i	\(0.207534\pi\)
							0.128037	+	0.991769i	\(0.459132\pi\)
\(11\)	407.934	+	196.450i	0.0924092	+	0.0445019i	0.479517	−	0.877532i	\(-0.340812\pi\)
							−0.387108	+	0.922034i	\(0.626526\pi\)
\(13\)	−2632.60	+	6707.76i	−0.332341	+	0.846791i	0.662842	+	0.748759i	\(0.269350\pi\)
							−0.995183	+	0.0980317i	\(0.968745\pi\)
\(17\)	3941.19	+	594.039i	0.194561	+	0.0293254i	0.245600	−	0.969371i	\(-0.421015\pi\)
							−0.0510386	+	0.998697i	\(0.516253\pi\)
\(19\)	3252.89	−	43406.9i	0.108801	−	1.45185i	−0.629303	−	0.777160i	\(-0.716660\pi\)
							0.738104	−	0.674687i	\(-0.235721\pi\)
\(23\)	−24223.7	+	16515.4i	−0.415138	+	0.283037i	−0.752823	−	0.658223i	\(-0.771309\pi\)
							0.337685	+	0.941259i	\(0.390356\pi\)
\(29\)	−150438.	+	139586.i	−1.14542	+	1.06280i	−0.148150	+	0.988965i	\(0.547332\pi\)
							−0.997271	+	0.0738304i	\(0.976478\pi\)
\(31\)	134196.	+	41393.9i	0.809046	+	0.249558i	0.671564	−	0.740947i	\(-0.265623\pi\)
							0.137482	+	0.990504i	\(0.456099\pi\)
\(37\)	−25872.3	+	44812.1i	−0.0839708	+	0.145442i	−0.904952	−	0.425513i	\(-0.860094\pi\)
							0.820981	+	0.570955i	\(0.193427\pi\)
\(41\)	9222.63	+	40407.0i	0.0208983	+	0.0915615i	0.984302	−	0.176495i	\(-0.0564758\pi\)
							−0.963403	+	0.268056i	\(0.913619\pi\)
\(43\)	273482.	−	443876.i	0.524553	−	0.851378i
							\(47\)	534680.	−	257488.i	0.751193	−	0.361756i	−0.0187869	−	0.999824i	\(-0.505980\pi\)
0.769980	+	0.638068i	\(0.220266\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.g.a.10.22	✓	300
43.13	even	21	inner	43.8.g.a.13.22	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.g.a.10.22	✓	300	1.1	even	1	trivial
43.8.g.a.13.22	yes	300	43.13	even	21	inner