Embedded newform 43.10.e.a.4.19

• Label: 43.10.e.a.4.19
• 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			4.19
Character	\(\chi\)	\(=\)	43.4
Dual form			43.10.e.a.11.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.12806 - 5.17643i) q^{2} +(85.9143 + 107.733i) q^{3} +(104.176 + 456.426i) q^{4} +(1027.10 + 494.625i) q^{5} +912.333 q^{6} +8194.69 q^{7} +(5846.90 + 2815.72i) q^{8} +(154.716 - 677.855i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 23 q^{2} + 141 q^{3} - 7771 q^{4} + 677 q^{5} - 6572 q^{6} - 23872 q^{7} + 40269 q^{8} - 322935 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.12806	−	5.17643i	0.182436	−	0.228768i	−0.682201	−	0.731165i	\(-0.738977\pi\)
							0.864637	+	0.502397i	\(0.167548\pi\)
\(3\)	85.9143	+	107.733i	0.612379	+	0.767899i	0.987250	−	0.159178i	\(-0.0508845\pi\)
							−0.374871	+	0.927077i	\(0.622313\pi\)
\(5\)	1027.10	+	494.625i	0.734933	+	0.353925i	0.763622	−	0.645663i	\(-0.223419\pi\)
							−0.0286893	+	0.999588i	\(0.509133\pi\)
\(7\)	8194.69			1.29001			0.645003	−	0.764180i	\(-0.276856\pi\)
							0.645003	+	0.764180i	\(0.276856\pi\)
\(11\)	11183.4	−	48997.7i	0.230307	−	1.00904i	−0.719079	−	0.694928i	\(-0.755436\pi\)
							0.949386	−	0.314112i	\(-0.101707\pi\)
\(13\)	129752.	+	62485.4i	1.26000	+	0.606783i	0.940174	−	0.340694i	\(-0.110662\pi\)
							0.319824	+	0.947477i	\(0.396376\pi\)
\(17\)	−296402.	+	142740.i	−0.860717	+	0.414500i	−0.811544	−	0.584291i	\(-0.801373\pi\)
							−0.0491727	+	0.998790i	\(0.515658\pi\)
\(19\)	−124835.	−	546940.i	−0.219759	−	0.962827i	−0.957656	−	0.287914i	\(-0.907038\pi\)
							0.737897	−	0.674913i	\(-0.235819\pi\)
\(23\)	−4853.41	+	21264.2i	−0.00361636	+	0.0158443i	−0.976704	−	0.214591i	\(-0.931158\pi\)
							0.973088	+	0.230436i	\(0.0740152\pi\)
\(29\)	−3.89693e6	+	4.88660e6i	−1.02313	+	1.28297i	−0.0646215	+	0.997910i	\(0.520584\pi\)
							−0.958511	+	0.285057i	\(0.907987\pi\)
\(31\)	3.24445e6	−	4.06841e6i	0.630977	−	0.791220i	−0.358866	−	0.933389i	\(-0.616836\pi\)
							0.989842	+	0.142169i	\(0.0454078\pi\)
\(37\)	−1.49998e6			−0.131577			−0.0657883	−	0.997834i	\(-0.520956\pi\)
							−0.0657883	+	0.997834i	\(0.520956\pi\)
\(41\)	−4.47600e6	+	5.61273e6i	−0.247379	+	0.310204i	−0.889982	−	0.455996i	\(-0.849283\pi\)
							0.642603	+	0.766199i	\(0.277855\pi\)
\(43\)	−1.35080e7	+	1.78921e7i	−0.602534	+	0.798093i
							\(47\)	−1.11887e7	−	4.90207e7i	−0.334455	−	1.46534i	−0.810405	−	0.585870i	\(-0.800753\pi\)
0.475950	−	0.879472i	\(-0.342104\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.e.a.4.19	✓	192
43.11	even	7	inner	43.10.e.a.11.19	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.e.a.4.19	✓	192	1.1	even	1	trivial
43.10.e.a.11.19	yes	192	43.11	even	7	inner