Embedded newform 43.6.g.a.9.8

• Label: 43.6.g.a.9.8
• 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			9.8
Character	\(\chi\)	\(=\)	43.9
Dual form			43.6.g.a.24.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23023 - 1.54266i) q^{2} +(-24.9627 - 3.76252i) q^{3} +(6.25434 - 27.4020i) q^{4} +(-22.2688 - 15.1826i) q^{5} +(24.9055 + 43.1377i) q^{6} +(-4.12718 + 7.14849i) q^{7} +(-106.854 + 51.4581i) q^{8} +(376.774 + 116.219i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - 468 q^{4} - 85 q^{5} - 22 q^{6} + 454 q^{7} + 698 q^{8} - 333 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.23023	−	1.54266i	−0.217476	−	0.272706i	0.661111	−	0.750288i	\(-0.270085\pi\)
							−0.878588	+	0.477581i	\(0.841514\pi\)
\(3\)	−24.9627	−	3.76252i	−1.60136	−	0.241366i	−0.713218	−	0.700943i	\(-0.752763\pi\)
							−0.888138	+	0.459577i	\(0.848001\pi\)
\(5\)	−22.2688	−	15.1826i	−0.398356	−	0.271595i	0.347529	−	0.937669i	\(-0.387021\pi\)
							−0.745885	+	0.666075i	\(0.767973\pi\)
\(7\)	−4.12718	+	7.14849i	−0.0318353	+	0.0551403i	−0.881504	−	0.472176i	\(-0.843469\pi\)
							0.849669	+	0.527317i	\(0.176802\pi\)
\(11\)	71.2864	+	312.326i	0.177634	+	0.778263i	0.982719	+	0.185104i	\(0.0592623\pi\)
							−0.805085	+	0.593159i	\(0.797881\pi\)
\(13\)	−30.7775	+	410.697i	−0.0505097	+	0.674005i	0.913278	+	0.407336i	\(0.133542\pi\)
							−0.963788	+	0.266669i	\(0.914077\pi\)
\(17\)	156.495	−	106.696i	0.131334	−	0.0895420i	−0.495871	−	0.868396i	\(-0.665151\pi\)
							0.627205	+	0.778854i	\(0.284199\pi\)
\(19\)	1640.35	−	505.982i	1.04245	−	0.321552i	0.274177	−	0.961679i	\(-0.411595\pi\)
							0.768269	+	0.640127i	\(0.221118\pi\)
\(23\)	−2402.07	+	2228.80i	−0.946819	+	0.878520i	−0.992722	−	0.120426i	\(-0.961574\pi\)
							0.0459034	+	0.998946i	\(0.485383\pi\)
\(29\)	−477.370	+	71.9520i	−0.105405	+	0.0158872i	−0.201533	−	0.979482i	\(-0.564592\pi\)
							0.0961280	+	0.995369i	\(0.469354\pi\)
\(31\)	−2361.91	+	6018.06i	−0.441428	+	1.12474i	0.521085	+	0.853505i	\(0.325527\pi\)
							−0.962513	+	0.271236i	\(0.912568\pi\)
\(37\)	−583.042	−	1009.86i	−0.0700157	−	0.121271i	0.828892	−	0.559408i	\(-0.188972\pi\)
							−0.898908	+	0.438138i	\(0.855638\pi\)
\(41\)	−5332.59	−	6686.85i	−0.495426	−	0.621244i	0.469765	−	0.882791i	\(-0.344339\pi\)
							−0.965191	+	0.261547i	\(0.915767\pi\)
\(43\)	−12012.8	−	1643.76i	−0.990768	−	0.135572i
							\(47\)	−5959.71	+	26111.2i	−0.393533	+	1.72418i	0.258521	+	0.966006i	\(0.416765\pi\)
−0.652053	+	0.758173i	\(0.726092\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.g.a.9.8	✓	204
43.24	even	21	inner	43.6.g.a.24.8	yes	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.g.a.9.8	✓	204	1.1	even	1	trivial
43.6.g.a.24.8	yes	204	43.24	even	21	inner