Embedded newform 43.4.e.a.4.2

• Label: 43.4.e.a.4.2
• 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			4.2
Character	\(\chi\)	\(=\)	43.4
Dual form			43.4.e.a.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.32023 + 2.90947i) q^{2} +(3.61377 + 4.53152i) q^{3} +(-1.30141 - 5.70184i) q^{4} +(13.3469 + 6.42753i) q^{5} -21.5691 q^{6} -12.2090 q^{7} +(-7.21369 - 3.47393i) q^{8} +(-1.46730 + 6.42866i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 9 q^{2} - 3 q^{3} - 31 q^{4} - 23 q^{5} + 16 q^{6} + 96 q^{7} + 61 q^{8} - 177 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.32023	+	2.90947i	−0.820324	+	1.02865i	0.178675	+	0.983908i	\(0.442819\pi\)
							−0.998998	+	0.0447453i	\(0.985752\pi\)
\(3\)	3.61377	+	4.53152i	0.695470	+	0.872091i	0.996676	−	0.0814675i	\(-0.0259607\pi\)
							−0.301206	+	0.953559i	\(0.597389\pi\)
\(5\)	13.3469	+	6.42753i	1.19378	+	0.574896i	0.921897	−	0.387434i	\(-0.126639\pi\)
							0.271886	+	0.962330i	\(0.412353\pi\)
\(7\)	−12.2090			−0.659222			−0.329611	−	0.944117i	\(-0.606918\pi\)
							−0.329611	+	0.944117i	\(0.606918\pi\)
\(11\)	−0.456732	+	2.00108i	−0.0125191	+	0.0548497i	−0.980802	−	0.195008i	\(-0.937527\pi\)
							0.968283	+	0.249858i	\(0.0803839\pi\)
\(13\)	−8.58668	−	4.13513i	−0.183194	−	0.0882214i	0.340039	−	0.940411i	\(-0.389560\pi\)
							−0.523232	+	0.852190i	\(0.675274\pi\)
\(17\)	66.7975	−	32.1680i	0.952987	−	0.458934i	0.108254	−	0.994123i	\(-0.465474\pi\)
							0.844732	+	0.535189i	\(0.179760\pi\)
\(19\)	22.9418	+	100.515i	0.277011	+	1.21367i	0.901551	+	0.432673i	\(0.142429\pi\)
							−0.624540	+	0.780993i	\(0.714713\pi\)
\(23\)	32.7067	−	143.298i	0.296514	−	1.29911i	−0.578764	−	0.815495i	\(-0.696465\pi\)
							0.875279	−	0.483619i	\(-0.160678\pi\)
\(29\)	−63.7295	+	79.9142i	−0.408078	+	0.511714i	−0.942820	−	0.333302i	\(-0.891837\pi\)
							0.534742	+	0.845015i	\(0.320409\pi\)
\(31\)	−192.150	+	240.949i	−1.11326	+	1.39599i	−0.204404	+	0.978887i	\(0.565526\pi\)
							−0.908860	+	0.417102i	\(0.863046\pi\)
\(37\)	434.968			1.93266			0.966329	−	0.257312i	\(-0.0828367\pi\)
							0.966329	+	0.257312i	\(0.0828367\pi\)
\(41\)	114.686	−	143.812i	0.436853	−	0.547797i	−0.513857	−	0.857876i	\(-0.671784\pi\)
							0.950711	+	0.310079i	\(0.100355\pi\)
\(43\)	−177.188	−	219.343i	−0.628393	−	0.777896i
							\(47\)	−77.5868	−	339.930i	−0.240791	−	1.05498i	−0.940299	−	0.340351i	\(-0.889454\pi\)
0.699507	−	0.714626i	\(-0.253403\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.e.a.4.2	✓	60
43.11	even	7	inner	43.4.e.a.11.2	yes	60
43.21	even	7		1849.4.a.h.1.6		30
43.22	odd	14		1849.4.a.g.1.25		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.e.a.4.2	✓	60	1.1	even	1	trivial
43.4.e.a.11.2	yes	60	43.11	even	7	inner
1849.4.a.g.1.25		30	43.22	odd	14
1849.4.a.h.1.6		30	43.21	even	7