Embedded newform 43.8.c.a.6.16

• Label: 43.8.c.a.6.16
• Level: $43$
• Weight: $8$
• Character: 43.6
• Analytic conductor: $13.433$
• Analytic rank: $0$
• Dimension: $50$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	43.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.4325560958\)
Analytic rank:	\(0\)
Dimension:	\(50\)
Relative dimension:	\(25\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			6.16
Character	\(\chi\)	\(=\)	43.6
Dual form			43.8.c.a.36.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+5.12015 q^{2} +(23.6605 - 40.9812i) q^{3} -101.784 q^{4} +(246.931 - 427.696i) q^{5} +(121.145 - 209.830i) q^{6} +(159.926 + 276.999i) q^{7} -1176.53 q^{8} +(-26.1400 - 45.2757i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 26 q^{2} - 2 q^{3} + 3014 q^{4} - 249 q^{5} - 1281 q^{6} - 687 q^{7} + 5496 q^{8} - 13389 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.12015			0.452562			0.226281	−	0.974062i	\(-0.427343\pi\)
							0.226281	+	0.974062i	\(0.427343\pi\)
\(3\)	23.6605	−	40.9812i	0.505941	−	0.876315i	−0.494035	−	0.869442i	\(-0.664479\pi\)
							0.999976	−	0.00687362i	\(-0.00218796\pi\)
\(5\)	246.931	−	427.696i	0.883446	−	1.53017i	0.0359607	−	0.999353i	\(-0.488551\pi\)
							0.847485	−	0.530819i	\(-0.178116\pi\)
\(7\)	159.926	+	276.999i	0.176228	+	0.305236i	0.940586	−	0.339557i	\(-0.110277\pi\)
							−0.764357	+	0.644793i	\(0.776944\pi\)
\(11\)	−4644.18			−1.05205			−0.526023	−	0.850470i	\(-0.676317\pi\)
							−0.526023	+	0.850470i	\(0.676317\pi\)
\(13\)	−3894.64	−	6745.71i	−0.491660	−	0.851581i	0.508294	−	0.861184i	\(-0.330277\pi\)
							−0.999954	+	0.00960316i	\(0.996943\pi\)
\(17\)	−1197.61	−	2074.32i	−0.0591212	−	0.102401i	0.834950	−	0.550326i	\(-0.185496\pi\)
							−0.894071	+	0.447925i	\(0.852163\pi\)
\(19\)	−5637.21	+	9763.93i	−0.188550	+	0.326578i	−0.944767	−	0.327743i	\(-0.893712\pi\)
							0.756217	+	0.654321i	\(0.227045\pi\)
\(23\)	27110.3	−	46956.4i	0.464608	−	0.804724i	−0.534576	−	0.845120i	\(-0.679529\pi\)
							0.999184	+	0.0403960i	\(0.0128620\pi\)
\(29\)	−83225.3	−	144151.i	−0.633669	−	1.09755i	−0.986795	−	0.161972i	\(-0.948215\pi\)
							0.353126	−	0.935576i	\(-0.385119\pi\)
\(31\)	−56263.7	+	97451.5i	−0.339205	+	0.587520i	−0.984283	−	0.176596i	\(-0.943491\pi\)
							0.645079	+	0.764116i	\(0.276825\pi\)
\(37\)	−55211.8	+	95629.7i	−0.179195	+	0.310375i	−0.941605	−	0.336719i	\(-0.890683\pi\)
							0.762410	+	0.647094i	\(0.224016\pi\)
\(41\)	791795.			1.79419			0.897097	−	0.441833i	\(-0.145672\pi\)
							0.897097	+	0.441833i	\(0.145672\pi\)
\(43\)	−74285.3	+	516043.i	−0.142483	+	0.989797i
							\(47\)	84232.6			0.118342			0.0591709	−	0.998248i	\(-0.481154\pi\)
0.0591709	+	0.998248i	\(0.481154\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.c.a.6.16	✓	50
43.36	even	3	inner	43.8.c.a.36.16	yes	50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.c.a.6.16	✓	50	1.1	even	1	trivial
43.8.c.a.36.16	yes	50	43.36	even	3	inner