Embedded newform 43.8.e.a.4.19

• Label: 43.8.e.a.4.19
• Level: $43$
• Weight: $8$
• Character: 43.4
• Analytic conductor: $13.433$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	43.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.4325560958\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(24\) 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			43.8.e.a.11.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(9.61177 - 12.0528i) q^{2} +(6.35304 + 7.96646i) q^{3} +(-24.4006 - 106.906i) q^{4} +(89.1946 + 42.9539i) q^{5} +157.082 q^{6} +751.914 q^{7} +(254.795 + 122.703i) q^{8} +(463.550 - 2030.94i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q + q^{2} - 33 q^{3} - 1531 q^{4} - 253 q^{5} + 484 q^{6} + 3440 q^{7} - 6339 q^{8} - 7461 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	9.61177	−	12.0528i	0.849568	−	1.06532i	−0.147519	−	0.989059i	\(-0.547129\pi\)
							0.997088	−	0.0762656i	\(-0.0242997\pi\)
\(3\)	6.35304	+	7.96646i	0.135849	+	0.170349i	0.845103	−	0.534604i	\(-0.179539\pi\)
							−0.709253	+	0.704954i	\(0.750968\pi\)
\(5\)	89.1946	+	42.9539i	0.319112	+	0.153676i	0.586582	−	0.809890i	\(-0.300473\pi\)
							−0.267470	+	0.963566i	\(0.586187\pi\)
\(7\)	751.914			0.828562			0.414281	−	0.910149i	\(-0.364033\pi\)
							0.414281	+	0.910149i	\(0.364033\pi\)
\(11\)	884.462	−	3875.08i	0.200357	−	0.877822i	−0.770362	−	0.637606i	\(-0.779925\pi\)
							0.970720	−	0.240216i	\(-0.0772181\pi\)
\(13\)	−4401.68	−	2119.74i	−0.555670	−	0.267597i	0.134903	−	0.990859i	\(-0.456928\pi\)
							−0.690574	+	0.723262i	\(0.742642\pi\)
\(17\)	2447.74	−	1178.77i	0.120835	−	0.0581912i	−0.372489	−	0.928036i	\(-0.621496\pi\)
							0.493325	+	0.869845i	\(0.335781\pi\)
\(19\)	7987.60	+	34995.9i	0.267164	+	1.17052i	0.913296	+	0.407297i	\(0.133529\pi\)
							−0.646131	+	0.763226i	\(0.723614\pi\)
\(23\)	3502.03	−	15343.4i	0.0600167	−	0.262950i	−0.936014	−	0.351963i	\(-0.885514\pi\)
							0.996030	+	0.0890130i	\(0.0283713\pi\)
\(29\)	−97452.9	+	122202.i	−0.741997	+	0.930434i	−0.999356	−	0.0358815i	\(-0.988576\pi\)
							0.257359	+	0.966316i	\(0.417148\pi\)
\(31\)	2225.15	−	2790.25i	0.0134151	−	0.0168220i	−0.775079	−	0.631865i	\(-0.782290\pi\)
							0.788494	+	0.615043i	\(0.210861\pi\)
\(37\)	456618.			1.48199			0.740997	−	0.671508i	\(-0.234353\pi\)
							0.740997	+	0.671508i	\(0.234353\pi\)
\(41\)	42230.5	−	52955.3i	0.0956935	−	0.119996i	−0.731679	−	0.681649i	\(-0.761263\pi\)
							0.827373	+	0.561653i	\(0.189835\pi\)
\(43\)	−33566.7	+	520281.i	−0.0643826	+	0.997925i
							\(47\)	202393.	+	886742.i	0.284350	+	1.24582i	0.892154	+	0.451732i	\(0.149194\pi\)
−0.607804	+	0.794087i	\(0.707949\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.e.a.4.19	✓	144
43.11	even	7	inner	43.8.e.a.11.19	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.e.a.4.19	✓	144	1.1	even	1	trivial
43.8.e.a.11.19	yes	144	43.11	even	7	inner