Embedded newform 43.8.c.a.6.6

• Label: 43.8.c.a.6.6
• 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			6.6
Character	\(\chi\)	\(=\)	43.6
Dual form			43.8.c.a.36.6

$q$-expansion

\(f(q)\)	\(=\)	\(q-13.8355 q^{2} +(-40.6999 + 70.4944i) q^{3} +63.4217 q^{4} +(-8.95282 + 15.5067i) q^{5} +(563.105 - 975.327i) q^{6} +(-449.623 - 778.770i) q^{7} +893.474 q^{8} +(-2219.47 - 3844.24i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 26 q^{2} - 2 q^{3} + 3014 q^{4} - 249 q^{5} - 1281 q^{6} - 687 q^{7} + 5496 q^{8} - 13389 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.8355			−1.22290			−0.611450	−	0.791283i	\(-0.709413\pi\)
							−0.611450	+	0.791283i	\(0.709413\pi\)
\(3\)	−40.6999	+	70.4944i	−0.870301	+	1.50741i	−0.00861527	+	0.999963i	\(0.502742\pi\)
							−0.861686	+	0.507442i	\(0.830591\pi\)
\(5\)	−8.95282	+	15.5067i	−0.0320306	+	0.0554786i	−0.881596	−	0.472004i	\(-0.843531\pi\)
							0.849566	+	0.527483i	\(0.176864\pi\)
\(7\)	−449.623	−	778.770i	−0.495457	−	0.858156i	0.504530	−	0.863394i	\(-0.331666\pi\)
							−0.999986	+	0.00523831i	\(0.998333\pi\)
\(11\)	−959.458			−0.217346			−0.108673	−	0.994078i	\(-0.534660\pi\)
							−0.108673	+	0.994078i	\(0.534660\pi\)
\(13\)	−1140.14	−	1974.78i	−0.143932	−	0.249297i	0.785042	−	0.619442i	\(-0.212641\pi\)
							−0.928974	+	0.370145i	\(0.879308\pi\)
\(17\)	5479.46	+	9490.71i	0.270500	+	0.468519i	0.968990	−	0.247100i	\(-0.0794777\pi\)
							−0.698490	+	0.715620i	\(0.746144\pi\)
\(19\)	12528.4	−	21699.9i	0.419044	−	0.725805i	−0.576800	−	0.816886i	\(-0.695699\pi\)
							0.995844	+	0.0910803i	\(0.0290320\pi\)
\(23\)	−5683.50	+	9844.12i	−0.0974022	+	0.168706i	−0.910609	−	0.413270i	\(-0.864387\pi\)
							0.813206	+	0.581975i	\(0.197720\pi\)
\(29\)	−70018.1	−	121275.i	−0.533110	−	0.923374i	−0.999252	−	0.0386642i	\(-0.987690\pi\)
							0.466142	−	0.884710i	\(-0.345644\pi\)
\(31\)	−51081.5	+	88475.7i	−0.307962	+	0.533406i	−0.977916	−	0.208996i	\(-0.932980\pi\)
							0.669954	+	0.742402i	\(0.266314\pi\)
\(37\)	−255651.	+	442801.i	−0.829740	+	1.43715i	0.0685029	+	0.997651i	\(0.478178\pi\)
							−0.898243	+	0.439500i	\(0.855156\pi\)
\(41\)	−386481.			−0.875760			−0.437880	−	0.899034i	\(-0.644270\pi\)
							−0.437880	+	0.899034i	\(0.644270\pi\)
\(43\)	15952.1	−	521118.i	0.0305970	−	0.999532i
							\(47\)	1.16322e6			1.63425			0.817124	−	0.576461i	\(-0.195567\pi\)
0.817124	+	0.576461i	\(0.195567\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.c.a.6.6	✓	50
43.36	even	3	inner	43.8.c.a.36.6	yes	50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.c.a.6.6	✓	50	1.1	even	1	trivial
43.8.c.a.36.6	yes	50	43.36	even	3	inner