Embedded newform 43.10.e.a.4.3

• Label: 43.10.e.a.4.3
• Level: $43$
• Weight: $10$
• Character: 43.4
• Analytic conductor: $22.147$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	43.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(22.1465409550\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(32\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			4.3
Character	\(\chi\)	\(=\)	43.4
Dual form			43.10.e.a.11.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-26.1677 + 32.8132i) q^{2} +(147.714 + 185.228i) q^{3} +(-278.030 - 1218.13i) q^{4} +(460.421 + 221.727i) q^{5} -9943.26 q^{6} -4982.64 q^{7} +(27885.8 + 13429.1i) q^{8} +(-8109.94 + 35532.0i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 23 q^{2} + 141 q^{3} - 7771 q^{4} + 677 q^{5} - 6572 q^{6} - 23872 q^{7} + 40269 q^{8} - 322935 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−26.1677	+	32.8132i	−1.15646	+	1.45015i	−0.285786	+	0.958294i	\(0.592255\pi\)
							−0.870674	+	0.491861i	\(0.836317\pi\)
\(3\)	147.714	+	185.228i	1.05287	+	1.32026i	0.945349	+	0.326060i	\(0.105721\pi\)
							0.107525	+	0.994202i	\(0.465708\pi\)
\(5\)	460.421	+	221.727i	0.329451	+	0.158655i	0.591296	−	0.806455i	\(-0.298617\pi\)
							−0.261845	+	0.965110i	\(0.584331\pi\)
\(7\)	−4982.64			−0.784365			−0.392182	−	0.919887i	\(-0.628280\pi\)
							−0.392182	+	0.919887i	\(0.628280\pi\)
\(11\)	18600.3	−	81493.1i	0.383047	−	1.67824i	−0.304827	−	0.952408i	\(-0.598599\pi\)
							0.687874	−	0.725830i	\(-0.258544\pi\)
\(13\)	−119894.	−	57737.7i	−1.16426	−	0.560680i	−0.250975	−	0.967994i	\(-0.580751\pi\)
							−0.913288	+	0.407314i	\(0.866465\pi\)
\(17\)	169621.	−	81685.0i	0.492560	−	0.237204i	−0.171079	−	0.985257i	\(-0.554725\pi\)
							0.663639	+	0.748053i	\(0.269011\pi\)
\(19\)	−130783.	−	572998.i	−0.230229	−	1.00870i	−0.949450	−	0.313918i	\(-0.898358\pi\)
							0.719221	−	0.694781i	\(-0.244499\pi\)
\(23\)	356231.	−	1.56075e6i	0.265434	−	1.16294i	−0.649828	−	0.760081i	\(-0.725159\pi\)
							0.915262	−	0.402860i	\(-0.131984\pi\)
\(29\)	−1.92335e6	+	2.41181e6i	−0.504973	+	0.633216i	−0.967343	−	0.253471i	\(-0.918428\pi\)
							0.462370	+	0.886687i	\(0.346999\pi\)
\(31\)	−1.87925e6	+	2.35651e6i	−0.365475	+	0.458291i	−0.930235	−	0.366964i	\(-0.880397\pi\)
							0.564760	+	0.825255i	\(0.308969\pi\)
\(37\)	8.09648e6			0.710212			0.355106	−	0.934826i	\(-0.384445\pi\)
							0.355106	+	0.934826i	\(0.384445\pi\)
\(41\)	−1.84379e7	+	2.31204e7i	−1.01902	+	1.27781i	−0.0588897	+	0.998264i	\(0.518756\pi\)
							−0.960132	+	0.279548i	\(0.909815\pi\)
\(43\)	−2.23476e7	−	1.78267e6i	−0.996833	−	0.0795174i
							\(47\)	−4.31034e6	−	1.88848e7i	−0.128846	−	0.564511i	−0.997598	−	0.0692643i	\(-0.977935\pi\)
0.868752	−	0.495247i	\(-0.164922\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.e.a.4.3	✓	192
43.11	even	7	inner	43.10.e.a.11.3	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.e.a.4.3	✓	192	1.1	even	1	trivial
43.10.e.a.11.3	yes	192	43.11	even	7	inner