Embedded newform 43.6.g.a.9.16

• Label: 43.6.g.a.9.16
• Level: $43$
• Weight: $6$
• Character: 43.9
• Analytic conductor: $6.897$
• Analytic rank: $0$
• Dimension: $204$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.89650425196\)
Analytic rank:	\(0\)
Dimension:	\(204\)
Relative dimension:	\(17\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			9.16
Character	\(\chi\)	\(=\)	43.9
Dual form			43.6.g.a.24.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.80897 + 7.28422i) q^{2} +(-11.9998 - 1.80868i) q^{3} +(-12.1950 + 53.4298i) q^{4} +(-56.0782 - 38.2335i) q^{5} +(-56.5318 - 97.9160i) q^{6} +(-58.6801 + 101.637i) q^{7} +(-191.420 + 92.1833i) q^{8} +(-91.4795 - 28.2177i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - 468 q^{4} - 85 q^{5} - 22 q^{6} + 454 q^{7} + 698 q^{8} - 333 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.80897	+	7.28422i	1.02689	+	1.28768i	0.956989	+	0.290126i	\(0.0936971\pi\)
							0.0699015	+	0.997554i	\(0.477731\pi\)
\(3\)	−11.9998	−	1.80868i	−0.769790	−	0.116027i	−0.247607	−	0.968861i	\(-0.579644\pi\)
							−0.522183	+	0.852833i	\(0.674882\pi\)
\(5\)	−56.0782	−	38.2335i	−1.00316	−	0.683941i	−0.0539338	−	0.998545i	\(-0.517176\pi\)
							−0.949223	+	0.314604i	\(0.898128\pi\)
\(7\)	−58.6801	+	101.637i	−0.452633	+	0.783983i	−0.998549	−	0.0538570i	\(-0.982848\pi\)
							0.545916	+	0.837840i	\(0.316182\pi\)
\(11\)	−31.3087	−	137.172i	−0.0780160	−	0.341810i	0.920823	−	0.389980i	\(-0.127518\pi\)
							−0.998839	+	0.0481701i	\(0.984661\pi\)
\(13\)	−65.8375	+	878.541i	−0.108048	+	1.44180i	0.634977	+	0.772531i	\(0.281010\pi\)
							−0.743024	+	0.669264i	\(0.766609\pi\)
\(17\)	160.523	−	109.443i	0.134715	−	0.0918472i	−0.494087	−	0.869412i	\(-0.664498\pi\)
							0.628803	+	0.777565i	\(0.283545\pi\)
\(19\)	529.140	−	163.218i	0.336269	−	0.103725i	−0.122022	−	0.992527i	\(-0.538938\pi\)
							0.458291	+	0.888802i	\(0.348462\pi\)
\(23\)	−3306.81	+	3068.27i	−1.30343	+	1.20941i	−0.340322	+	0.940309i	\(0.610536\pi\)
							−0.963112	+	0.269101i	\(0.913273\pi\)
\(29\)	−5209.62	+	785.224i	−1.15030	+	0.173380i	−0.696387	−	0.717666i	\(-0.745210\pi\)
							−0.453913	+	0.891046i	\(0.649972\pi\)
\(31\)	1007.03	−	2565.88i	0.188208	−	0.479547i	−0.805359	−	0.592787i	\(-0.798028\pi\)
							0.993568	+	0.113240i	\(0.0361228\pi\)
\(37\)	667.512	+	1156.17i	0.0801595	+	0.138840i	0.903318	−	0.428971i	\(-0.141124\pi\)
							−0.823159	+	0.567811i	\(0.807790\pi\)
\(41\)	96.5327	+	121.048i	0.00896840	+	0.0112460i	0.786295	−	0.617851i	\(-0.211996\pi\)
							−0.777327	+	0.629097i	\(0.783425\pi\)
\(43\)	7856.77	+	9234.70i	0.647997	+	0.761643i
							\(47\)	6086.59	−	26667.1i	0.401910	−	1.76088i	−0.217740	−	0.976007i	\(-0.569869\pi\)
0.619651	−	0.784878i	\(-0.287274\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.g.a.9.16	✓	204
43.24	even	21	inner	43.6.g.a.24.16	yes	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.g.a.9.16	✓	204	1.1	even	1	trivial
43.6.g.a.24.16	yes	204	43.24	even	21	inner