Embedded newform 43.4.e.a.4.9

• Label: 43.4.e.a.4.9
• Level: $43$
• Weight: $4$
• Character: 43.4
• Analytic conductor: $2.537$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	43.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.53708213025\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(10\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			4.9
Character	\(\chi\)	\(=\)	43.4
Dual form			43.4.e.a.11.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.72209 - 3.41339i) q^{2} +(-2.84196 - 3.56370i) q^{3} +(-2.46131 - 10.7837i) q^{4} +(8.30189 + 3.99798i) q^{5} -19.9004 q^{6} -12.2688 q^{7} +(-12.0406 - 5.79846i) q^{8} +(1.38482 - 6.06730i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 9 q^{2} - 3 q^{3} - 31 q^{4} - 23 q^{5} + 16 q^{6} + 96 q^{7} + 61 q^{8} - 177 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.72209	−	3.41339i	0.962404	−	1.20682i	−0.0159488	−	0.999873i	\(-0.505077\pi\)
							0.978353	−	0.206944i	\(-0.0663517\pi\)
\(3\)	−2.84196	−	3.56370i	−0.546935	−	0.685834i	0.429148	−	0.903234i	\(-0.358814\pi\)
							−0.976083	+	0.217400i	\(0.930242\pi\)
\(5\)	8.30189	+	3.99798i	0.742544	+	0.357590i	0.766604	−	0.642120i	\(-0.221945\pi\)
							−0.0240601	+	0.999711i	\(0.507659\pi\)
\(7\)	−12.2688			−0.662455			−0.331228	−	0.943551i	\(-0.607463\pi\)
							−0.331228	+	0.943551i	\(0.607463\pi\)
\(11\)	−3.74140	+	16.3922i	−0.102552	+	0.449311i	0.897415	+	0.441188i	\(0.145443\pi\)
							−0.999967	+	0.00812292i	\(0.997414\pi\)
\(13\)	66.8370	+	32.1870i	1.42594	+	0.686697i	0.978238	−	0.207485i	\(-0.0665278\pi\)
							0.447703	+	0.894182i	\(0.352242\pi\)
\(17\)	−0.640842	+	0.308613i	−0.00914277	+	0.00440293i	−0.438450	−	0.898756i	\(-0.644472\pi\)
							0.429307	+	0.903159i	\(0.358758\pi\)
\(19\)	−6.96294	−	30.5066i	−0.0840741	−	0.368353i	0.915336	−	0.402691i	\(-0.131925\pi\)
							−0.999410	+	0.0343378i	\(0.989068\pi\)
\(23\)	−18.6609	+	81.7588i	−0.169177	+	0.741213i	0.817152	+	0.576423i	\(0.195552\pi\)
							−0.986329	+	0.164790i	\(0.947305\pi\)
\(29\)	−142.592	+	178.805i	−0.913060	+	1.14494i	0.0759532	+	0.997111i	\(0.475800\pi\)
							−0.989013	+	0.147829i	\(0.952771\pi\)
\(31\)	−151.938	+	190.525i	−0.880289	+	1.10385i	0.113607	+	0.993526i	\(0.463759\pi\)
							−0.993896	+	0.110321i	\(0.964812\pi\)
\(37\)	82.4344			0.366274			0.183137	−	0.983087i	\(-0.441375\pi\)
							0.183137	+	0.983087i	\(0.441375\pi\)
\(41\)	252.021	−	316.025i	0.959979	−	1.20378i	−0.0190013	−	0.999819i	\(-0.506049\pi\)
							0.978980	−	0.203956i	\(-0.0653799\pi\)
\(43\)	−163.858	+	229.473i	−0.581118	+	0.813820i
							\(47\)	−78.0998	−	342.177i	−0.242383	−	1.06195i	−0.938840	−	0.344353i	\(-0.888098\pi\)
0.696457	−	0.717599i	\(-0.254759\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.e.a.4.9	✓	60
43.11	even	7	inner	43.4.e.a.11.9	yes	60
43.21	even	7		1849.4.a.h.1.25		30
43.22	odd	14		1849.4.a.g.1.6		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.e.a.4.9	✓	60	1.1	even	1	trivial
43.4.e.a.11.9	yes	60	43.11	even	7	inner
1849.4.a.g.1.6		30	43.22	odd	14
1849.4.a.h.1.25		30	43.21	even	7