Embedded newform 43.8.c.a.6.24

• Label: 43.8.c.a.6.24
• 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			6.24
Character	\(\chi\)	\(=\)	43.6
Dual form			43.8.c.a.36.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+21.0873 q^{2} +(0.504502 - 0.873824i) q^{3} +316.675 q^{4} +(-262.689 + 454.990i) q^{5} +(10.6386 - 18.4266i) q^{6} +(58.1593 + 100.735i) q^{7} +3978.65 q^{8} +(1092.99 + 1893.12i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 26 q^{2} - 2 q^{3} + 3014 q^{4} - 249 q^{5} - 1281 q^{6} - 687 q^{7} + 5496 q^{8} - 13389 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	21.0873			1.86387			0.931937	−	0.362621i	\(-0.118118\pi\)
							0.931937	+	0.362621i	\(0.118118\pi\)
\(3\)	0.504502	−	0.873824i	0.0107879	−	0.0186853i	−0.860581	−	0.509314i	\(-0.829899\pi\)
							0.871369	+	0.490628i	\(0.163233\pi\)
\(5\)	−262.689	+	454.990i	−0.939824	+	1.62782i	−0.174027	+	0.984741i	\(0.555678\pi\)
							−0.765797	+	0.643082i	\(0.777655\pi\)
\(7\)	58.1593	+	100.735i	0.0640879	+	0.111004i	0.896289	−	0.443470i	\(-0.146253\pi\)
							−0.832201	+	0.554474i	\(0.812920\pi\)
\(11\)	4731.97			1.07193			0.535966	−	0.844240i	\(-0.319948\pi\)
							0.535966	+	0.844240i	\(0.319948\pi\)
\(13\)	−5481.01	−	9493.38i	−0.691924	−	1.19845i	−0.971207	−	0.238239i	\(-0.923430\pi\)
							0.279282	−	0.960209i	\(-0.409903\pi\)
\(17\)	−5786.65	−	10022.8i	−0.285664	−	0.494785i	0.687106	−	0.726557i	\(-0.258881\pi\)
							−0.972770	+	0.231773i	\(0.925547\pi\)
\(19\)	21145.5	−	36625.0i	0.707261	−	1.22501i	−0.258609	−	0.965982i	\(-0.583264\pi\)
							0.965869	−	0.259029i	\(-0.0834027\pi\)
\(23\)	−10537.6	+	18251.7i	−0.180591	+	0.312793i	−0.942082	−	0.335383i	\(-0.891134\pi\)
							0.761491	+	0.648176i	\(0.224468\pi\)
\(29\)	47634.6	+	82505.6i	0.362685	+	0.628189i	0.988402	−	0.151862i	\(-0.0485268\pi\)
							−0.625717	+	0.780050i	\(0.715193\pi\)
\(31\)	−130185.	+	225487.i	−0.784867	+	1.35943i	0.144212	+	0.989547i	\(0.453935\pi\)
							−0.929079	+	0.369882i	\(0.879398\pi\)
\(37\)	20018.1	−	34672.3i	0.0649705	−	0.112532i	−0.831710	−	0.555210i	\(-0.812638\pi\)
							0.896681	+	0.442677i	\(0.145971\pi\)
\(41\)	407417.			0.923201			0.461600	−	0.887088i	\(-0.347275\pi\)
							0.461600	+	0.887088i	\(0.347275\pi\)
\(43\)	−27643.5	−	520629.i	−0.0530217	−	0.998593i
							\(47\)	−85716.9			−0.120427			−0.0602135	−	0.998186i	\(-0.519178\pi\)
−0.0602135	+	0.998186i	\(0.519178\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.c.a.6.24	✓	50
43.36	even	3	inner	43.8.c.a.36.24	yes	50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.c.a.6.24	✓	50	1.1	even	1	trivial
43.8.c.a.36.24	yes	50	43.36	even	3	inner