Embedded newform 43.8.g.a.10.10

• Label: 43.8.g.a.10.10
• 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			10.10
Character	\(\chi\)	\(=\)	43.10
Dual form			43.8.g.a.13.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41715 - 6.20892i) q^{2} +(-9.22088 - 8.55573i) q^{3} +(78.7816 - 37.9392i) q^{4} +(-154.133 + 23.2318i) q^{5} +(-40.0545 + 69.3764i) q^{6} +(79.0979 + 137.002i) q^{7} +(-855.463 - 1072.72i) q^{8} +(-151.611 - 2023.10i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 40 q^{2} - 12 q^{3} - 3028 q^{4} + 235 q^{5} + 1274 q^{6} - 4122 q^{7} + 762 q^{8} + 8097 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41715	−	6.20892i	−0.125259	−	0.548796i	−0.998146	−	0.0608728i	\(-0.980612\pi\)
							0.872886	−	0.487924i	\(-0.162246\pi\)
\(3\)	−9.22088	−	8.55573i	−0.197173	−	0.182950i	0.575407	−	0.817867i	\(-0.304844\pi\)
							−0.772580	+	0.634917i	\(0.781034\pi\)
\(5\)	−154.133	+	23.2318i	−0.551442	+	0.0831165i	−0.418852	−	0.908054i	\(-0.637567\pi\)
							−0.132590	+	0.991171i	\(0.542329\pi\)
\(7\)	79.0979	+	137.002i	0.0871609	+	0.150967i	0.906310	−	0.422614i	\(-0.138887\pi\)
							−0.819149	+	0.573581i	\(0.805554\pi\)
\(11\)	−392.666	−	189.098i	−0.0889507	−	0.0428364i	0.388879	−	0.921289i	\(-0.372862\pi\)
							−0.477830	+	0.878452i	\(0.658577\pi\)
\(13\)	−1714.57	+	4368.65i	−0.216448	+	0.551500i	−0.997318	−	0.0731939i	\(-0.976681\pi\)
							0.780870	+	0.624693i	\(0.214776\pi\)
\(17\)	−32169.4	−	4848.75i	−1.58808	−	0.239364i	−0.705213	−	0.708995i	\(-0.749149\pi\)
							−0.882863	+	0.469631i	\(0.844387\pi\)
\(19\)	−557.720	+	7442.26i	−0.0186543	+	0.248924i	0.980079	+	0.198608i	\(0.0636420\pi\)
							−0.998733	+	0.0503164i	\(0.983977\pi\)
\(23\)	−81033.1	+	55247.4i	−1.38872	+	0.946813i	−0.389048	+	0.921218i	\(0.627196\pi\)
							−0.999672	+	0.0255960i	\(0.991852\pi\)
\(29\)	112132.	−	104043.i	0.853760	−	0.792174i	−0.126067	−	0.992022i	\(-0.540235\pi\)
							0.979827	+	0.199848i	\(0.0640448\pi\)
\(31\)	40556.3	+	12510.0i	0.244508	+	0.0754206i	0.414587	−	0.910010i	\(-0.363926\pi\)
							−0.170079	+	0.985430i	\(0.554402\pi\)
\(37\)	−67489.0	+	116894.i	−0.219042	+	0.379392i	−0.954515	−	0.298162i	\(-0.903626\pi\)
							0.735473	+	0.677554i	\(0.236960\pi\)
\(41\)	−118530.	−	519315.i	−0.268587	−	1.17676i	−0.911658	−	0.410951i	\(-0.865197\pi\)
							0.643070	−	0.765807i	\(-0.277660\pi\)
\(43\)	100148.	−	511653.i	0.192089	−	0.981378i
							\(47\)	1.07404e6	−	517231.i	1.50896	−	0.726678i	0.517332	−	0.855785i	\(-0.326925\pi\)
0.991631	+	0.129107i	\(0.0412111\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.g.a.10.10	✓	300
43.13	even	21	inner	43.8.g.a.13.10	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.g.a.10.10	✓	300	1.1	even	1	trivial
43.8.g.a.13.10	yes	300	43.13	even	21	inner