Embedded newform 43.8.c.a.6.17

• Label: 43.8.c.a.6.17
• 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			6.17
Character	\(\chi\)	\(=\)	43.6
Dual form			43.8.c.a.36.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+7.44937 q^{2} +(37.3503 - 64.6927i) q^{3} -72.5069 q^{4} +(-250.945 + 434.650i) q^{5} +(278.237 - 481.920i) q^{6} +(-47.3845 - 82.0723i) q^{7} -1493.65 q^{8} +(-1696.60 - 2938.59i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 26 q^{2} - 2 q^{3} + 3014 q^{4} - 249 q^{5} - 1281 q^{6} - 687 q^{7} + 5496 q^{8} - 13389 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.44937			0.658437			0.329219	−	0.944254i	\(-0.393215\pi\)
							0.329219	+	0.944254i	\(0.393215\pi\)
\(3\)	37.3503	−	64.6927i	0.798675	−	1.38335i	−0.121804	−	0.992554i	\(-0.538868\pi\)
							0.920479	−	0.390792i	\(-0.127799\pi\)
\(5\)	−250.945	+	434.650i	−0.897810	+	1.55505i	−0.0675215	+	0.997718i	\(0.521509\pi\)
							−0.830288	+	0.557334i	\(0.811824\pi\)
\(7\)	−47.3845	−	82.0723i	−0.0522147	−	0.0904386i	0.838737	−	0.544537i	\(-0.183295\pi\)
							−0.890951	+	0.454099i	\(0.849961\pi\)
\(11\)	−4866.78			−1.10247			−0.551236	−	0.834349i	\(-0.685844\pi\)
							−0.551236	+	0.834349i	\(0.685844\pi\)
\(13\)	−3383.59	−	5860.54i	−0.427145	−	0.739837i	0.569473	−	0.822010i	\(-0.307147\pi\)
							−0.996618	+	0.0821729i	\(0.973814\pi\)
\(17\)	9833.32	+	17031.8i	0.485433	+	0.840794i	0.999860	−	0.0167402i	\(-0.00532881\pi\)
							−0.514427	+	0.857534i	\(0.671995\pi\)
\(19\)	−4700.14	+	8140.89i	−0.157208	+	0.272291i	−0.933861	−	0.357637i	\(-0.883583\pi\)
							0.776653	+	0.629929i	\(0.216916\pi\)
\(23\)	−32078.6	+	55561.8i	−0.549754	+	0.952201i	0.448537	+	0.893764i	\(0.351945\pi\)
							−0.998291	+	0.0584373i	\(0.981388\pi\)
\(29\)	−50681.2	−	87782.3i	−0.385881	−	0.668366i	0.606010	−	0.795457i	\(-0.292769\pi\)
							−0.991891	+	0.127091i	\(0.959436\pi\)
\(31\)	81744.7	−	141586.i	0.492826	−	0.853600i	−0.507140	−	0.861864i	\(-0.669297\pi\)
							0.999966	+	0.00826385i	\(0.00263050\pi\)
\(37\)	211826.	−	366893.i	0.687500	−	1.19079i	−0.285144	−	0.958485i	\(-0.592041\pi\)
							0.972644	−	0.232300i	\(-0.0746253\pi\)
\(41\)	−453772.			−1.02824			−0.514119	−	0.857719i	\(-0.671881\pi\)
							−0.514119	+	0.857719i	\(0.671881\pi\)
\(43\)	−510159.	+	107503.i	−0.978511	+	0.206196i
							\(47\)	−247519.			−0.347749			−0.173874	−	0.984768i	\(-0.555629\pi\)
−0.173874	+	0.984768i	\(0.555629\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.c.a.6.17	✓	50
43.36	even	3	inner	43.8.c.a.36.17	yes	50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.c.a.6.17	✓	50	1.1	even	1	trivial
43.8.c.a.36.17	yes	50	43.36	even	3	inner