Embedded newform 43.8.c.a.6.4

• Label: 43.8.c.a.6.4
• 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			6.4
Character	\(\chi\)	\(=\)	43.6
Dual form			43.8.c.a.36.4

$q$-expansion

\(f(q)\)	\(=\)	\(q-17.3293 q^{2} +(-16.5117 + 28.5992i) q^{3} +172.305 q^{4} +(-197.750 + 342.513i) q^{5} +(286.137 - 495.604i) q^{6} +(401.179 + 694.863i) q^{7} -767.772 q^{8} +(548.224 + 949.553i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 26 q^{2} - 2 q^{3} + 3014 q^{4} - 249 q^{5} - 1281 q^{6} - 687 q^{7} + 5496 q^{8} - 13389 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−17.3293			−1.53171			−0.765854	−	0.643014i	\(-0.777684\pi\)
							−0.765854	+	0.643014i	\(0.777684\pi\)
\(3\)	−16.5117	+	28.5992i	−0.353076	+	0.611546i	−0.986787	−	0.162024i	\(-0.948198\pi\)
							0.633710	+	0.773570i	\(0.281531\pi\)
\(5\)	−197.750	+	342.513i	−0.707491	+	1.22541i	0.258294	+	0.966066i	\(0.416840\pi\)
							−0.965785	+	0.259344i	\(0.916494\pi\)
\(7\)	401.179	+	694.863i	0.442074	+	0.765695i	0.997843	−	0.0656415i	\(-0.0209094\pi\)
							−0.555769	+	0.831337i	\(0.687576\pi\)
\(11\)	7437.89			1.68491			0.842453	−	0.538770i	\(-0.181111\pi\)
							0.842453	+	0.538770i	\(0.181111\pi\)
\(13\)	3633.59	+	6293.56i	0.458706	+	0.794502i	0.998893	−	0.0470433i	\(-0.0149799\pi\)
							−0.540187	+	0.841545i	\(0.681647\pi\)
\(17\)	13510.2	+	23400.4i	0.666947	+	1.15519i	0.978754	+	0.205040i	\(0.0657325\pi\)
							−0.311807	+	0.950145i	\(0.600934\pi\)
\(19\)	−3180.35	+	5508.52i	−0.106374	+	0.184246i	−0.914299	−	0.405040i	\(-0.867258\pi\)
							0.807925	+	0.589286i	\(0.200591\pi\)
\(23\)	−28710.5	+	49728.1i	−0.492033	+	0.852226i	−0.999958	−	0.00917539i	\(-0.997079\pi\)
							0.507925	+	0.861401i	\(0.330413\pi\)
\(29\)	109659.	+	189935.i	0.834934	+	1.44615i	0.894084	+	0.447898i	\(0.147827\pi\)
							−0.0591508	+	0.998249i	\(0.518839\pi\)
\(31\)	114831.	−	198894.i	0.692300	−	1.19910i	−0.278782	−	0.960354i	\(-0.589931\pi\)
							0.971082	−	0.238745i	\(-0.0767359\pi\)
\(37\)	168599.	−	292023.i	0.547205	−	0.947787i	−0.451259	−	0.892393i	\(-0.649025\pi\)
							0.998465	−	0.0553943i	\(-0.0176416\pi\)
\(41\)	127147.			0.288113			0.144057	−	0.989569i	\(-0.453985\pi\)
							0.144057	+	0.989569i	\(0.453985\pi\)
\(43\)	−470945.	+	223672.i	−0.903298	+	0.429014i
							\(47\)	696862.			0.979048			0.489524	−	0.871990i	\(-0.337171\pi\)
0.489524	+	0.871990i	\(0.337171\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.c.a.6.4	✓	50
43.36	even	3	inner	43.8.c.a.36.4	yes	50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.c.a.6.4	✓	50	1.1	even	1	trivial
43.8.c.a.36.4	yes	50	43.36	even	3	inner